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Theorem aomclem8 37111
Description: Lemma for dfac11 37112. 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 2001 . . . . . . 7 ( = 𝑏 → (𝑖𝑖𝑏))
2 elequ2 2001 . . . . . . . 8 (𝑔 = 𝑐 → (𝑖𝑔𝑖𝑐))
32notbid 308 . . . . . . 7 (𝑔 = 𝑐 → (¬ 𝑖𝑔 ↔ ¬ 𝑖𝑐))
41, 3bi2anan9r 917 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → ((𝑖 ∧ ¬ 𝑖𝑔) ↔ (𝑖𝑏 ∧ ¬ 𝑖𝑐)))
5 elequ2 2001 . . . . . . . . 9 (𝑔 = 𝑐 → (𝑗𝑔𝑗𝑐))
6 elequ2 2001 . . . . . . . . 9 ( = 𝑏 → (𝑗𝑗𝑏))
75, 6bi2bian9 918 . . . . . . . 8 ((𝑔 = 𝑐 = 𝑏) → ((𝑗𝑔𝑗) ↔ (𝑗𝑐𝑗𝑏)))
87imbi2d 330 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))))
98ralbidv 2980 . . . . . 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 746 . . . . 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 3045 . . . 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 1994 . . . . . . 7 (𝑖 = 𝑑 → (𝑖𝑏𝑑𝑏))
13 elequ1 1994 . . . . . . . 8 (𝑖 = 𝑑 → (𝑖𝑐𝑑𝑐))
1413notbid 308 . . . . . . 7 (𝑖 = 𝑑 → (¬ 𝑖𝑐 ↔ ¬ 𝑑𝑐))
1512, 14anbi12d 746 . . . . . 6 (𝑖 = 𝑑 → ((𝑖𝑏 ∧ ¬ 𝑖𝑐) ↔ (𝑑𝑏 ∧ ¬ 𝑑𝑐)))
16 breq2 4617 . . . . . . . . 9 (𝑖 = 𝑑 → (𝑗(𝑒 dom 𝑒)𝑖𝑗(𝑒 dom 𝑒)𝑑))
1716imbi1d 331 . . . . . . . 8 (𝑖 = 𝑑 → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)) ↔ (𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏))))
1817ralbidv 2980 . . . . . . 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 4616 . . . . . . . . 9 (𝑗 = 𝑓 → (𝑗(𝑒 dom 𝑒)𝑑𝑓(𝑒 dom 𝑒)𝑑))
20 elequ1 1994 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑐𝑓𝑐))
21 elequ1 1994 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑏𝑓𝑏))
2220, 21bibi12d 335 . . . . . . . . 9 (𝑗 = 𝑓 → ((𝑗𝑐𝑗𝑏) ↔ (𝑓𝑐𝑓𝑏)))
2319, 22imbi12d 334 . . . . . . . 8 (𝑗 = 𝑓 → ((𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏)) ↔ (𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏))))
2423cbvralv 3159 . . . . . . 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 746 . . . . 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 3160 . . . 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 4684 . 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 2761 . . 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 2761 . . . 4 𝑔(𝑦𝑐)
32 nfcv 2761 . . . 4 𝑔(𝑅1‘dom 𝑒)
33 nfopab1 4681 . . . 4 𝑔{⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
3431, 32, 33nfsup 8301 . . 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 6148 . . . 4 (𝑔 = 𝑐 → (𝑦𝑔) = (𝑦𝑐))
3635supeq1d 8296 . . 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 4709 . 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 2761 . . . 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 6156 . . . 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 5311 . . . . . 6 (𝑔 = 𝑐 → ran 𝑔 = ran 𝑐)
4140difeq2d 3706 . . . . 5 (𝑔 = 𝑐 → ((𝑅1‘dom 𝑒) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑐))
4241fveq2d 6152 . . . 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 4709 . . 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 7415 . . 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 1840 . . 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 1840 . . 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 4707 . . . . . . . 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 7416 . . . . . . 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 5261 . . . . . 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 2761 . . . . . 6 𝑔{𝑐}
5250, 51nfima 5433 . . . . 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 4451 . . . 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 2761 . . . . . 6 𝑔{𝑏}
5550, 54nfima 5433 . . . . 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 4451 . . . 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 2773 . . 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 2761 . . . . . . . . 9 V
59 nfcv 2761 . . . . . . . . . . . 12 (𝑦𝑔)
60 nfcv 2761 . . . . . . . . . . . 12 (𝑅1‘dom 𝑒)
61 nfopab2 4682 . . . . . . . . . . . 12 {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
6259, 60, 61nfsup 8301 . . . . . . . . . . 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 4706 . . . . . . . . . 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 2761 . . . . . . . . . 10 ((𝑅1‘dom 𝑒) ∖ ran 𝑔)
6563, 64nffv 6155 . . . . . . . . 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 4706 . . . . . . . 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 7416 . . . . . . 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 5261 . . . . . 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 2761 . . . . . 6 {𝑐}
7068, 69nfima 5433 . . . . 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 4451 . . . 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 2761 . . . . . 6 {𝑏}
7368, 72nfima 5433 . . . . 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 4451 . . . 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 2773 . . 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 4158 . . . . . 6 (𝑔 = 𝑐 → {𝑔} = {𝑐})
7776imaeq2d 5425 . . . . 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 4445 . . . 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 4158 . . . . . 6 ( = 𝑏 → {} = {𝑏})
8079imaeq2d 5425 . . . . 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 4445 . . . 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 2688 . . . 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 494 . . 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 4683 . 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 6148 . . . . 5 (𝑔 = 𝑐 → (rank‘𝑔) = (rank‘𝑐))
86 fveq2 6148 . . . . 5 ( = 𝑏 → (rank‘) = (rank‘𝑏))
8785, 86breqan12d 4629 . . . 4 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) E (rank‘) ↔ (rank‘𝑐) E (rank‘𝑏)))
8885, 86eqeqan12d 2637 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) = (rank‘) ↔ (rank‘𝑐) = (rank‘𝑏)))
89 simpl 473 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → 𝑔 = 𝑐)
90 suceq 5749 . . . . . . . . 9 ((rank‘𝑔) = (rank‘𝑐) → suc (rank‘𝑔) = suc (rank‘𝑐))
9185, 90syl 17 . . . . . . . 8 (𝑔 = 𝑐 → suc (rank‘𝑔) = suc (rank‘𝑐))
9291adantr 481 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → suc (rank‘𝑔) = suc (rank‘𝑐))
9392fveq2d 6152 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → (𝑒‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑐)))
94 simpr 477 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → = 𝑏)
9589, 93, 94breq123d 4627 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → (𝑔(𝑒‘suc (rank‘𝑔))𝑐(𝑒‘suc (rank‘𝑐))𝑏))
9688, 95anbi12d 746 . . . 4 ((𝑔 = 𝑐 = 𝑏) → (((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))) ↔ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏)))
9787, 96orbi12d 745 . . 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 4684 . 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 2621 . 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 5284 . . . . . . 7 (𝑙 = 𝑒 → dom 𝑙 = dom 𝑒)
101100unieqd 4412 . . . . . . 7 (𝑙 = 𝑒 dom 𝑙 = dom 𝑒)
102100, 101eqeq12d 2636 . . . . . 6 (𝑙 = 𝑒 → (dom 𝑙 = dom 𝑙 ↔ dom 𝑒 = dom 𝑒))
103 fveq1 6147 . . . . . . . . . 10 (𝑙 = 𝑒 → (𝑙‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑔)))
104103breqd 4624 . . . . . . . . 9 (𝑙 = 𝑒 → (𝑔(𝑙‘suc (rank‘𝑔))𝑔(𝑒‘suc (rank‘𝑔))))
105104anbi2d 739 . . . . . . . 8 (𝑙 = 𝑒 → (((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))) ↔ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔)))))
106105orbi2d 737 . . . . . . 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 4678 . . . . . 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 2622 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑦𝑔) = (𝑦𝑔))
109100fveq2d 6152 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑅1‘dom 𝑙) = (𝑅1‘dom 𝑒))
110101fveq2d 6152 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑒 → (𝑅1 dom 𝑙) = (𝑅1 dom 𝑒))
111 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑒𝑙 = 𝑒)
112111, 101fveq12d 6154 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑒 → (𝑙 dom 𝑙) = (𝑒 dom 𝑒))
113112breqd 4624 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑒 → (𝑗(𝑙 dom 𝑙)𝑖𝑗(𝑒 dom 𝑒)𝑖))
114113imbi1d 331 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑒 → ((𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))))
115110, 114raleqbidv 3141 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑒 → (∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)) ↔ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))))
116115anbi2d 739 . . . . . . . . . . . . . . . . . 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 3142 . . . . . . . . . . . . . . . . 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 4678 . . . . . . . . . . . . . . . 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 8300 . . . . . . . . . . . . . . 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 4705 . . . . . . . . . . . . . 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 3705 . . . . . . . . . . . . . 14 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑔))
122120, 121fveq12d 6154 . . . . . . . . . . . . 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 4705 . . . . . . . . . . . 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 7415 . . . . . . . . . . . 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 5258 . . . . . . . . . 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 5424 . . . . . . . . 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 4445 . . . . . . . 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 5424 . . . . . . . . 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 4445 . . . . . . . 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 2692 . . . . . . 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 4678 . . . . . 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 4085 . . . . 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 5101 . . . . 5 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)) = ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))
135133, 134ineq12d 3793 . . . 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 4710 . . 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 7415 . . 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 2852 . . . . 5 (𝑎 = 𝑐 → (𝑎 ≠ ∅ ↔ 𝑐 ≠ ∅))
142 fveq2 6148 . . . . . 6 (𝑎 = 𝑐 → (𝑦𝑎) = (𝑦𝑐))
143 pweq 4133 . . . . . . . 8 (𝑎 = 𝑐 → 𝒫 𝑎 = 𝒫 𝑐)
144143ineq1d 3791 . . . . . . 7 (𝑎 = 𝑐 → (𝒫 𝑎 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
145144difeq1d 3705 . . . . . 6 (𝑎 = 𝑐 → ((𝒫 𝑎 ∩ Fin) ∖ {∅}) = ((𝒫 𝑐 ∩ Fin) ∖ {∅}))
146142, 145eleq12d 2692 . . . . 5 (𝑎 = 𝑐 → ((𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}) ↔ (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
147141, 146imbi12d 334 . . . 4 (𝑎 = 𝑐 → ((𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ (𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅}))))
148147cbvralv 3159 . . 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 37110 1 (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cin 3554  c0 3891  ifcif 4058  𝒫 cpw 4130  {csn 4148   cuni 4402   cint 4440   class class class wbr 4613  {copab 4672  cmpt 4673   E cep 4983   We wwe 5032   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  Oncon0 5682  suc csuc 5684  cfv 5847  recscrecs 7412  Fincfn 7899  supcsup 8290  𝑅1cr1 8569  rankcrnk 8570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-er 7687  df-map 7804  df-en 7900  df-fin 7903  df-sup 8292  df-r1 8571  df-rank 8572
This theorem is referenced by:  dfac11  37112
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