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Theorem aomclem8 43073
Description: Lemma for dfac11 43074. 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 2123 . . . . . . 7 ( = 𝑏 → (𝑖𝑖𝑏))
2 elequ2 2123 . . . . . . . 8 (𝑔 = 𝑐 → (𝑖𝑔𝑖𝑐))
32notbid 318 . . . . . . 7 (𝑔 = 𝑐 → (¬ 𝑖𝑔 ↔ ¬ 𝑖𝑐))
41, 3bi2anan9r 639 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → ((𝑖 ∧ ¬ 𝑖𝑔) ↔ (𝑖𝑏 ∧ ¬ 𝑖𝑐)))
5 elequ2 2123 . . . . . . . . 9 (𝑔 = 𝑐 → (𝑗𝑔𝑗𝑐))
6 elequ2 2123 . . . . . . . . 9 ( = 𝑏 → (𝑗𝑗𝑏))
75, 6bi2bian9 640 . . . . . . . 8 ((𝑔 = 𝑐 = 𝑏) → ((𝑗𝑔𝑗) ↔ (𝑗𝑐𝑗𝑏)))
87imbi2d 340 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏))))
98ralbidv 3178 . . . . . 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 632 . . . . 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 3179 . . . 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 2115 . . . . . . 7 (𝑖 = 𝑑 → (𝑖𝑏𝑑𝑏))
13 elequ1 2115 . . . . . . . 8 (𝑖 = 𝑑 → (𝑖𝑐𝑑𝑐))
1413notbid 318 . . . . . . 7 (𝑖 = 𝑑 → (¬ 𝑖𝑐 ↔ ¬ 𝑑𝑐))
1512, 14anbi12d 632 . . . . . 6 (𝑖 = 𝑑 → ((𝑖𝑏 ∧ ¬ 𝑖𝑐) ↔ (𝑑𝑏 ∧ ¬ 𝑑𝑐)))
16 breq2 5147 . . . . . . . . 9 (𝑖 = 𝑑 → (𝑗(𝑒 dom 𝑒)𝑖𝑗(𝑒 dom 𝑒)𝑑))
1716imbi1d 341 . . . . . . . 8 (𝑖 = 𝑑 → ((𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑐𝑗𝑏)) ↔ (𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏))))
1817ralbidv 3178 . . . . . . 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 5146 . . . . . . . . 9 (𝑗 = 𝑓 → (𝑗(𝑒 dom 𝑒)𝑑𝑓(𝑒 dom 𝑒)𝑑))
20 elequ1 2115 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑐𝑓𝑐))
21 elequ1 2115 . . . . . . . . . 10 (𝑗 = 𝑓 → (𝑗𝑏𝑓𝑏))
2220, 21bibi12d 345 . . . . . . . . 9 (𝑗 = 𝑓 → ((𝑗𝑐𝑗𝑏) ↔ (𝑓𝑐𝑓𝑏)))
2319, 22imbi12d 344 . . . . . . . 8 (𝑗 = 𝑓 → ((𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏)) ↔ (𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏))))
2423cbvralvw 3237 . . . . . . 7 (∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑑 → (𝑗𝑐𝑗𝑏)) ↔ ∀𝑓 ∈ (𝑅1 dom 𝑒)(𝑓(𝑒 dom 𝑒)𝑑 → (𝑓𝑐𝑓𝑏)))
2518, 24bitrdi 287 . . . . . 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 632 . . . . 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)))))
2726cbvrexvw 3238 . . . 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, 27bitrdi 287 . . 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 5216 . 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 2905 . . 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 2905 . . . 4 𝑔(𝑦𝑐)
32 nfcv 2905 . . . 4 𝑔(𝑅1‘dom 𝑒)
33 nfopab1 5213 . . . 4 𝑔{⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
3431, 32, 33nfsup 9491 . . 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 6906 . . . 4 (𝑔 = 𝑐 → (𝑦𝑔) = (𝑦𝑐))
3635supeq1d 9486 . . 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 5253 . 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 2905 . . . 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 6917 . . . 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 5947 . . . . . 6 (𝑔 = 𝑐 → ran 𝑔 = ran 𝑐)
4140difeq2d 4126 . . . . 5 (𝑔 = 𝑐 → ((𝑅1‘dom 𝑒) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑐))
4241fveq2d 6910 . . . 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 5253 . . 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 8414 . . 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 1914 . . 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 1914 . . 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 5250 . . . . . . . 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 8415 . . . . . . 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 5889 . . . . . 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 2905 . . . . . 6 𝑔{𝑐}
5250, 51nfima 6086 . . . . 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 4956 . . . 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 2905 . . . . . 6 𝑔{𝑏}
5550, 54nfima 6086 . . . . 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 4956 . . . 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 2920 . . 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 2905 . . . . . . . . 9 V
59 nfcv 2905 . . . . . . . . . . . 12 (𝑦𝑔)
60 nfcv 2905 . . . . . . . . . . . 12 (𝑅1‘dom 𝑒)
61 nfopab2 5214 . . . . . . . . . . . 12 {⟨𝑔, ⟩ ∣ ∃𝑖 ∈ (𝑅1 dom 𝑒)((𝑖 ∧ ¬ 𝑖𝑔) ∧ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗)))}
6259, 60, 61nfsup 9491 . . . . . . . . . . 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 5249 . . . . . . . . . 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 2905 . . . . . . . . . 10 ((𝑅1‘dom 𝑒) ∖ ran 𝑔)
6563, 64nffv 6916 . . . . . . . . 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 5249 . . . . . . . 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 8415 . . . . . . 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 5889 . . . . . 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 2905 . . . . . 6 {𝑐}
7068, 69nfima 6086 . . . . 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 4956 . . . 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 2905 . . . . . 6 {𝑏}
7368, 72nfima 6086 . . . . 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 4956 . . . 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 2920 . . 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 4636 . . . . . 6 (𝑔 = 𝑐 → {𝑔} = {𝑐})
7776imaeq2d 6078 . . . . 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 4951 . . . 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 4636 . . . . . 6 ( = 𝑏 → {} = {𝑏})
8079imaeq2d 6078 . . . . 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 4951 . . . 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 2831 . . . 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 596 . . 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 5215 . 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 6906 . . . . 5 (𝑔 = 𝑐 → (rank‘𝑔) = (rank‘𝑐))
86 fveq2 6906 . . . . 5 ( = 𝑏 → (rank‘) = (rank‘𝑏))
8785, 86breqan12d 5159 . . . 4 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) E (rank‘) ↔ (rank‘𝑐) E (rank‘𝑏)))
8885, 86eqeqan12d 2751 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → ((rank‘𝑔) = (rank‘) ↔ (rank‘𝑐) = (rank‘𝑏)))
89 simpl 482 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → 𝑔 = 𝑐)
90 suceq 6450 . . . . . . . . 9 ((rank‘𝑔) = (rank‘𝑐) → suc (rank‘𝑔) = suc (rank‘𝑐))
9185, 90syl 17 . . . . . . . 8 (𝑔 = 𝑐 → suc (rank‘𝑔) = suc (rank‘𝑐))
9291adantr 480 . . . . . . 7 ((𝑔 = 𝑐 = 𝑏) → suc (rank‘𝑔) = suc (rank‘𝑐))
9392fveq2d 6910 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → (𝑒‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑐)))
94 simpr 484 . . . . . 6 ((𝑔 = 𝑐 = 𝑏) → = 𝑏)
9589, 93, 94breq123d 5157 . . . . 5 ((𝑔 = 𝑐 = 𝑏) → (𝑔(𝑒‘suc (rank‘𝑔))𝑐(𝑒‘suc (rank‘𝑐))𝑏))
9688, 95anbi12d 632 . . . 4 ((𝑔 = 𝑐 = 𝑏) → (((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔))) ↔ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏)))
9787, 96orbi12d 919 . . 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 5216 . 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 2737 . 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 5914 . . . . . . 7 (𝑙 = 𝑒 → dom 𝑙 = dom 𝑒)
101100unieqd 4920 . . . . . . 7 (𝑙 = 𝑒 dom 𝑙 = dom 𝑒)
102100, 101eqeq12d 2753 . . . . . 6 (𝑙 = 𝑒 → (dom 𝑙 = dom 𝑙 ↔ dom 𝑒 = dom 𝑒))
103 fveq1 6905 . . . . . . . . . 10 (𝑙 = 𝑒 → (𝑙‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑔)))
104103breqd 5154 . . . . . . . . 9 (𝑙 = 𝑒 → (𝑔(𝑙‘suc (rank‘𝑔))𝑔(𝑒‘suc (rank‘𝑔))))
105104anbi2d 630 . . . . . . . 8 (𝑙 = 𝑒 → (((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑙‘suc (rank‘𝑔))) ↔ ((rank‘𝑔) = (rank‘) ∧ 𝑔(𝑒‘suc (rank‘𝑔)))))
106105orbi2d 916 . . . . . . 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 5209 . . . . . 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 2738 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑦𝑔) = (𝑦𝑔))
109100fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑒 → (𝑅1‘dom 𝑙) = (𝑅1‘dom 𝑒))
110101fveq2d 6910 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑒 → (𝑅1 dom 𝑙) = (𝑅1 dom 𝑒))
111 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑒𝑙 = 𝑒)
112111, 101fveq12d 6913 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑒 → (𝑙 dom 𝑙) = (𝑒 dom 𝑒))
113112breqd 5154 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑒 → (𝑗(𝑙 dom 𝑙)𝑖𝑗(𝑒 dom 𝑒)𝑖))
114113imbi1d 341 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑒 → ((𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)) ↔ (𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))))
115110, 114raleqbidv 3346 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑒 → (∀𝑗 ∈ (𝑅1 dom 𝑙)(𝑗(𝑙 dom 𝑙)𝑖 → (𝑗𝑔𝑗)) ↔ ∀𝑗 ∈ (𝑅1 dom 𝑒)(𝑗(𝑒 dom 𝑒)𝑖 → (𝑗𝑔𝑗))))
116115anbi2d 630 . . . . . . . . . . . . . . . . . 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 3347 . . . . . . . . . . . . . . . . 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 5209 . . . . . . . . . . . . . . . 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 9490 . . . . . . . . . . . . . . 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 5244 . . . . . . . . . . . . . 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 4125 . . . . . . . . . . . . . 14 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑔))
122120, 121fveq12d 6913 . . . . . . . . . . . . 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 5244 . . . . . . . . . . . 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 8414 . . . . . . . . . . . 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 5886 . . . . . . . . . 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 6077 . . . . . . . . 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 4951 . . . . . . . 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 6077 . . . . . . . . 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 4951 . . . . . . . 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 2835 . . . . . . 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 5209 . . . . . 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 4554 . . . . 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 5717 . . . . 5 (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)) = ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))
135133, 134ineq12d 4221 . . . 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 5255 . . 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 8414 . . 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 3003 . . . . 5 (𝑎 = 𝑐 → (𝑎 ≠ ∅ ↔ 𝑐 ≠ ∅))
142 fveq2 6906 . . . . . 6 (𝑎 = 𝑐 → (𝑦𝑎) = (𝑦𝑐))
143 pweq 4614 . . . . . . . 8 (𝑎 = 𝑐 → 𝒫 𝑎 = 𝒫 𝑐)
144143ineq1d 4219 . . . . . . 7 (𝑎 = 𝑐 → (𝒫 𝑎 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
145144difeq1d 4125 . . . . . 6 (𝑎 = 𝑐 → ((𝒫 𝑎 ∩ Fin) ∖ {∅}) = ((𝒫 𝑐 ∩ Fin) ∖ {∅}))
146142, 145eleq12d 2835 . . . . 5 (𝑎 = 𝑐 → ((𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}) ↔ (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
147141, 146imbi12d 344 . . . 4 (𝑎 = 𝑐 → ((𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ (𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅}))))
148147cbvralvw 3237 . . 3 (∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ ∀𝑐 ∈ 𝒫 (𝑅1𝐴)(𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
149140, 148sylib 218 . 2 (𝜑 → ∀𝑐 ∈ 𝒫 (𝑅1𝐴)(𝑐 ≠ ∅ → (𝑦𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
15029, 37, 45, 84, 98, 99, 138, 139, 149aomclem7 43072 1 (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cin 3950  c0 4333  ifcif 4525  𝒫 cpw 4600  {csn 4626   cuni 4907   cint 4946   class class class wbr 5143  {copab 5205  cmpt 5225   E cep 5583   We wwe 5636   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  Oncon0 6384  suc csuc 6386  cfv 6561  recscrecs 8410  Fincfn 8985  supcsup 9480  𝑅1cr1 9802  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-map 8868  df-en 8986  df-fin 8989  df-sup 9482  df-r1 9804  df-rank 9805
This theorem is referenced by:  dfac11  43074
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