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Theorem aomclem8 38151
Description: Lemma for dfac11 38152. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem8.a (𝜑𝐴 ∈ On)
aomclem8.y (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
Assertion
Ref Expression
aomclem8 (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))
Distinct variable groups:   𝜑,𝑏   𝐴,𝑎,𝑏   𝑦,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑦,𝑎)   𝐴(𝑦)

Proof of Theorem aomclem8
Dummy variables 𝑐 𝑑 𝑒 𝑓 𝑔 𝑖 𝑗 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elequ2 2153 . . . . . . 7 ( = 𝑏 → (𝑖𝑖𝑏))
2 elequ2 2153 . . . . . . . 8 (𝑔 = 𝑐 → (𝑖𝑔𝑖𝑐))
32notbid 307 . . . . . . 7 (𝑔 = 𝑐 → (¬ 𝑖𝑔 ↔ ¬ 𝑖𝑐))
41, 3bi2anan9r 954 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → ((𝑖 ∧ ¬ 𝑖𝑔) ↔ (𝑖𝑏 ∧ ¬ 𝑖𝑐)))
5 elequ2 2153 . . . . . . . . 9 (𝑔 = 𝑐 → (𝑗𝑔𝑗𝑐))
6 elequ2 2153 . . . . . . . . 9 ( = 𝑏 → (𝑗𝑗𝑏))
75, 6bi2bian9 955 . . . . . . . 8 ((𝑔 = 𝑐 = 𝑏) → ((𝑗𝑔𝑗) ↔ (𝑗𝑐𝑗𝑏)))
87imbi2d 329 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))))
98ralbidv 3124 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)) ↔ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))))
104, 9anbi12d 749 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → (((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))) ↔ ((𝑖𝑏 ∧ ¬ 𝑖𝑐) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)))))
1110rexbidv 3190 . . . 4 ((𝑔 = 𝑐 = 𝑏) → (∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))) ↔ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖𝑏 ∧ ¬ 𝑖𝑐) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)))))
12 elequ1 2146 . . . . . . 7 (𝑖 = 𝑑 → (𝑖𝑏𝑑𝑏))
13 elequ1 2146 . . . . . . . 8 (𝑖 = 𝑑 → (𝑖𝑐𝑑𝑐))
1413notbid 307 . . . . . . 7 (𝑖 = 𝑑 → (¬ 𝑖𝑐 ↔ ¬ 𝑑𝑐))
1512, 14anbi12d 749 . . . . . 6 (𝑖 = 𝑑 → ((𝑖𝑏 ∧ ¬ 𝑖𝑐) ↔ (𝑑𝑏 ∧ ¬ 𝑑𝑐)))
16 breq2 4808 . . . . . . . . 9 (𝑖 = 𝑑 → (𝑗(𝑒 dom 𝑒)𝑖𝑗(𝑒 dom 𝑒)𝑑))
1716imbi1d 330 . . . . . . . 8 (𝑖 = 𝑑 → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)) ↔ (𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏))))
1817ralbidv 3124 . . . . . . 7 (𝑖 = 𝑑 → (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)) ↔ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏))))
19 breq1 4807 . . . . . . . . 9 (𝑗 = 𝑓 → (𝑗(𝑒 dom 𝑒)𝑑𝑓(𝑒 dom 𝑒)𝑑))
20 elequ1 2146 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑐𝑓𝑐))
21 elequ1 2146 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑏𝑓𝑏))
2220, 21bibi12d 334 . . . . . . . . 9 (𝑗 = 𝑓 → ((𝑗𝑐𝑗𝑏) ↔ (𝑓𝑐𝑓𝑏)))
2319, 22imbi12d 333 . . . . . . . 8 (𝑗 = 𝑓 → ((𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏)) ↔ (𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏))))
2423cbvralv 3310 . . . . . . 7 (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏)) ↔ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))
2518, 24syl6bb 276 . . . . . 6 (𝑖 = 𝑑 → (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)) ↔ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏))))
2615, 25anbi12d 749 . . . . 5 (𝑖 = 𝑑 → (((𝑖𝑏 ∧ ¬ 𝑖𝑐) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))) ↔ ((𝑑𝑏 ∧ ¬ 𝑑𝑐) ∧ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))))
2726cbvrexv 3311 . . . 4 (∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖𝑏 ∧ ¬ 𝑖𝑐) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))) ↔ ∃𝑑 ∈ (𝑅1 dom 𝑒)((𝑑𝑏 ∧ ¬ 𝑑𝑐) ∧ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏))))
2811, 27syl6bb 276 . . 3 ((𝑔 = 𝑐 = 𝑏) → (∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))) ↔ ∃𝑑 ∈ (𝑅1 dom 𝑒)((𝑑𝑏 ∧ ¬ 𝑑𝑐) ∧ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))))
2928cbvopabv 4874 . 2 {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))} = {⟨𝑐, 𝑏⟩ ∣ ∃𝑑 ∈ (𝑅1 dom 𝑒)((𝑑𝑏 ∧ ¬ 𝑑𝑐) ∧ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))}
30 nfcv 2902 . . 3 𝑐sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})
31 nfcv 2902 . . . 4 𝑔(𝑦𝑐)
32 nfcv 2902 . . . 4 𝑔(𝑅1‘dom 𝑒)
33 nfopab1 4871 . . . 4 𝑔{⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
3431, 32, 33nfsup 8524 . . 3 𝑔sup((𝑦𝑐), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})
35 fveq2 6353 . . . 4 (𝑔 = 𝑐 → (𝑦𝑔) = (𝑦𝑐))
3635supeq1d 8519 . . 3 (𝑔 = 𝑐 → sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}) = sup((𝑦𝑐), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))
3730, 34, 36cbvmpt 4901 . 2 (𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})) = (𝑐 ∈ V ↦ sup((𝑦𝑐), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))
38 nfcv 2902 . . . 4 𝑐((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))
39 nffvmpt1 6361 . . . 4 𝑔((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐))
40 rneq 5506 . . . . . 6 (𝑔 = 𝑐 → ran 𝑔 = ran 𝑐)
4140difeq2d 3871 . . . . 5 (𝑔 = 𝑐 → ((𝑅1‘dom 𝑒) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑐))
4241fveq2d 6357 . . . 4 (𝑔 = 𝑐 → ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)) = ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐)))
4338, 39, 42cbvmpt 4901 . . 3 (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))) = (𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐)))
44 recseq 7640 . . 3 ((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))) = (𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐))) → recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) = recs((𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐)))))
4543, 44ax-mp 5 . 2 recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) = recs((𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐))))
46 nfv 1992 . . 3 𝑐 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})
47 nfv 1992 . . 3 𝑏 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})
48 nfmpt1 4899 . . . . . . . 8 𝑔(𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))
4948nfrecs 7641 . . . . . . 7 𝑔recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
5049nfcnv 5456 . . . . . 6 𝑔recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
51 nfcv 2902 . . . . . 6 𝑔{𝑐}
5250, 51nfima 5632 . . . . 5 𝑔(recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
5352nfint 4638 . . . 4 𝑔 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
54 nfcv 2902 . . . . . 6 𝑔{𝑏}
5550, 54nfima 5632 . . . . 5 𝑔(recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
5655nfint 4638 . . . 4 𝑔 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
5753, 56nfel 2915 . . 3 𝑔 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
58 nfcv 2902 . . . . . . . . 9 V
59 nfcv 2902 . . . . . . . . . . . 12 (𝑦𝑔)
60 nfcv 2902 . . . . . . . . . . . 12 (𝑅1‘dom 𝑒)
61 nfopab2 4872 . . . . . . . . . . . 12 {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
6259, 60, 61nfsup 8524 . . . . . . . . . . 11 sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})
6358, 62nfmpt 4898 . . . . . . . . . 10 (𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))
64 nfcv 2902 . . . . . . . . . 10 ((𝑅1‘dom 𝑒) ∖ ran 𝑔)
6563, 64nffv 6360 . . . . . . . . 9 ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))
6658, 65nfmpt 4898 . . . . . . . 8 (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))
6766nfrecs 7641 . . . . . . 7 recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
6867nfcnv 5456 . . . . . 6 recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
69 nfcv 2902 . . . . . 6 {𝑐}
7068, 69nfima 5632 . . . . 5 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
7170nfint 4638 . . . 4 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
72 nfcv 2902 . . . . . 6 {𝑏}
7368, 72nfima 5632 . . . . 5 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
7473nfint 4638 . . . 4 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
7571, 74nfel 2915 . . 3 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
76 sneq 4331 . . . . . 6 (𝑔 = 𝑐 → {𝑔} = {𝑐})
7776imaeq2d 5624 . . . . 5 (𝑔 = 𝑐 → (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}))
7877inteqd 4632 . . . 4 (𝑔 = 𝑐 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}))
79 sneq 4331 . . . . . 6 ( = 𝑏 → {} = {𝑏})
8079imaeq2d 5624 . . . . 5 ( = 𝑏 → (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏}))
8180inteqd 4632 . . . 4 ( = 𝑏 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏}))
82 eleq12 2829 . . . 4 (( (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∧ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})) → ( (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) ↔ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})))
8378, 81, 82syl2an 495 . . 3 ((𝑔 = 𝑐 = 𝑏) → ( (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}) ↔ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})))
8446, 47, 57, 75, 83cbvopab 4873 . 2 {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})} = {⟨𝑐, 𝑏⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})}
85 fveq2 6353 . . . . 5 (𝑔 = 𝑐 → (rank‘𝑔) = (rank‘𝑐))
86 fveq2 6353 . . . . 5 ( = 𝑏 → (rank‘) = (rank‘𝑏))
8785, 86breqan12d 4820 . . . 4 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) E (rank‘) ↔ (rank‘𝑐) E (rank‘𝑏)))
8885, 86eqeqan12d 2776 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) = (rank‘) ↔ (rank‘𝑐) = (rank‘𝑏)))
89 simpl 474 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → 𝑔 = 𝑐)
90 suceq 5951 . . . . . . . . 9 ((rank‘𝑔) = (rank‘𝑐) → suc (rank‘𝑔) = suc (rank‘𝑐))
9185, 90syl 17 . . . . . . . 8 (𝑔 = 𝑐 → suc (rank‘𝑔) = suc (rank‘𝑐))
9291adantr 472 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → suc (rank‘𝑔) = suc (rank‘𝑐))
9392fveq2d 6357 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → (𝑒‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑐)))
94 simpr 479 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → = 𝑏)
9589, 93, 94breq123d 4818 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → (𝑔(𝑒‘suc (rank‘𝑔))𝑐(𝑒‘suc (rank‘𝑐))𝑏))
9688, 95anbi12d 749 . . . 4 ((𝑔 = 𝑐 = 𝑏) → (((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))) ↔ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏)))
9787, 96orbi12d 748 . . 3 ((𝑔 = 𝑐 = 𝑏) → (((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔)))) ↔ ((rank‘𝑐) E (rank‘𝑏) ∨ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏))))
9897cbvopabv 4874 . 2 {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))} = {⟨𝑐, 𝑏⟩ ∣ ((rank‘𝑐) E (rank‘𝑏) ∨ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏))}
99 eqid 2760 . 2 (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))) = (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))
100 dmeq 5479 . . . . . . 7 (𝑙 = 𝑒 → dom 𝑙 = dom 𝑒)
101100unieqd 4598 . . . . . . 7 (𝑙 = 𝑒 dom 𝑙 = dom 𝑒)
102100, 101eqeq12d 2775 . . . . . 6 (𝑙 = 𝑒 → (dom 𝑙 = dom 𝑙 ↔ dom 𝑒 = dom 𝑒))
103 fveq1 6352 . . . . . . . . . 10 (𝑙 = 𝑒 → (𝑙‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑔)))
104103breqd 4815 . . . . . . . . 9 (𝑙 = 𝑒 → (𝑔(𝑙‘suc (rank‘𝑔))𝑔(𝑒‘suc (rank‘𝑔))))
105104anbi2d 742 . . . . . . . 8 (𝑙 = 𝑒 → (((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))) ↔ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔)))))
106105orbi2d 740 . . . . . . 7 (𝑙 = 𝑒 → (((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔)))) ↔ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))))
107106opabbidv 4868 . . . . . 6 (𝑙 = 𝑒 → {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))} = {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))})
108 eqidd 2761 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑦𝑔) = (𝑦𝑔))
109100fveq2d 6357 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑅1‘dom 𝑙) = (𝑅1‘dom 𝑒))
110101fveq2d 6357 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑒 → (𝑅1 dom 𝑙) = (𝑅1 dom 𝑒))
111 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑒𝑙 = 𝑒)
112111, 101fveq12d 6359 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑒 → (𝑙 dom 𝑙) = (𝑒 dom 𝑒))
113112breqd 4815 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑒 → (𝑗(𝑙 dom 𝑙)𝑖𝑗(𝑒 dom 𝑒)𝑖))
114113imbi1d 330 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑒 → ((𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))))
115110, 114raleqbidv 3291 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑒 → (∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)) ↔ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))))
116115anbi2d 742 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑒 → (((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗))) ↔ ((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))))
117110, 116rexeqbidv 3292 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑒 → (∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗))) ↔ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))))
118117opabbidv 4868 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))} = {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})
119108, 109, 118supeq123d 8523 . . . . . . . . . . . . . . 15 (𝑙 = 𝑒 → sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}) = sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))
120119mpteq2dv 4897 . . . . . . . . . . . . . 14 (𝑙 = 𝑒 → (𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))})) = (𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))})))
121109difeq1d 3870 . . . . . . . . . . . . . 14 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑔))
122120, 121fveq12d 6359 . . . . . . . . . . . . 13 (𝑙 = 𝑒 → ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)) = ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))
123122mpteq2dv 4897 . . . . . . . . . . . 12 (𝑙 = 𝑒 → (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔))) = (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
124 recseq 7640 . . . . . . . . . . . 12 ((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔))) = (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))) → recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) = recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))))
125123, 124syl 17 . . . . . . . . . . 11 (𝑙 = 𝑒 → recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) = recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))))
126125cnveqd 5453 . . . . . . . . . 10 (𝑙 = 𝑒recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) = recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))))
127126imaeq1d 5623 . . . . . . . . 9 (𝑙 = 𝑒 → (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}))
128127inteqd 4632 . . . . . . . 8 (𝑙 = 𝑒 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}))
129126imaeq1d 5623 . . . . . . . . 9 (𝑙 = 𝑒 → (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}))
130129inteqd 4632 . . . . . . . 8 (𝑙 = 𝑒 (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {}) = (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {}))
131128, 130eleq12d 2833 . . . . . . 7 (𝑙 = 𝑒 → ( (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {}) ↔ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})))
132131opabbidv 4868 . . . . . 6 (𝑙 = 𝑒 → {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})} = {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})})
133102, 107, 132ifbieq12d 4257 . . . . 5 (𝑙 = 𝑒 → if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) = if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}))
134109sqxpeqd 5298 . . . . 5 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)) = ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))
135133, 134ineq12d 3958 . . . 4 (𝑙 = 𝑒 → (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙))) = (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))))
136135cbvmptv 4902 . . 3 (𝑙 ∈ V ↦ (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)))) = (𝑒 ∈ V ↦ (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))))
137 recseq 7640 . . 3 ((𝑙 ∈ V ↦ (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)))) = (𝑒 ∈ V ↦ (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))) → recs((𝑙 ∈ V ↦ (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙))))) = recs((𝑒 ∈ V ↦ (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))))))
138136, 137ax-mp 5 . 2 recs((𝑙 ∈ V ↦ (if(dom 𝑙 = dom 𝑙, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑙)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙))))) = recs((𝑒 ∈ V ↦ (if(dom 𝑒 = dom 𝑒, {⟨𝑔, ⟩ ∣ ((rank‘𝑔) E (rank‘) ∨ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))))}, {⟨𝑔, ⟩ ∣ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ (recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))))
139 aomclem8.a . 2 (𝜑𝐴 ∈ On)
140 aomclem8.y . . 3 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
141 neeq1 2994 . . . . 5 (𝑎 = 𝑐 → (𝑎 ≠ ∅ ↔ 𝑐 ≠ ∅))
142 fveq2 6353 . . . . . 6 (𝑎 = 𝑐 → (𝑦𝑎) = (𝑦𝑐))
143 pweq 4305 . . . . . . . 8 (𝑎 = 𝑐 → 𝒫 𝑎 = 𝒫 𝑐)
144143ineq1d 3956 . . . . . . 7 (𝑎 = 𝑐 → (𝒫 𝑎 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
145144difeq1d 3870 . . . . . 6 (𝑎 = 𝑐 → ((𝒫 𝑎 ∩ Fin) ∖ {∅}) = ((𝒫 𝑐 ∩ Fin) ∖ {∅}))
146142, 145eleq12d 2833 . . . . 5 (𝑎 = 𝑐 → ((𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}) ↔ (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
147141, 146imbi12d 333 . . . 4 (𝑎 = 𝑐 → ((𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ (𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅}))))
148147cbvralv 3310 . . 3 (∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ ∀𝑐 ∈ 𝒫 (𝑅1𝐴)(𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
149140, 148sylib 208 . 2 (𝜑 → ∀𝑐 ∈ 𝒫 (𝑅1𝐴)(𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
15029, 37, 45, 84, 98, 99, 138, 139, 149aomclem7 38150 1 (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wex 1853  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  cin 3714  c0 4058  ifcif 4230  𝒫 cpw 4302  {csn 4321   cuni 4588   cint 4627   class class class wbr 4804  {copab 4864  cmpt 4881   E cep 5178   We wwe 5224   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  cima 5269  Oncon0 5884  suc csuc 5886  cfv 6049  recscrecs 7637  Fincfn 8123  supcsup 8513  𝑅1cr1 8800  rankcrnk 8801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-er 7913  df-map 8027  df-en 8124  df-fin 8127  df-sup 8515  df-r1 8802  df-rank 8803
This theorem is referenced by:  dfac11  38152
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