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Theorem wfrlem17 7661
Description: Without using ax-rep 4957, show that all restrictions of wrecs are sets. (Contributed by Scott Fenton, 31-Jul-2020.)
Hypotheses
Ref Expression
wfrlem17.1 𝑅 We 𝐴
wfrlem17.2 𝑅 Se 𝐴
wfrlem17.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfrlem17 (𝑋 ∈ dom 𝐹 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)

Proof of Theorem wfrlem17
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem17.1 . . . . 5 𝑅 We 𝐴
2 wfrlem17.2 . . . . 5 𝑅 Se 𝐴
3 wfrlem17.3 . . . . 5 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
41, 2, 3wfrfun 7655 . . . 4 Fun 𝐹
5 funfvop 6545 . . . 4 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹)
64, 5mpan 673 . . 3 (𝑋 ∈ dom 𝐹 → ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹)
7 df-wrecs 7636 . . . . . 6 wrecs(𝑅, 𝐴, 𝐺) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
83, 7eqtri 2824 . . . . 5 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
98eleq2i 2873 . . . 4 (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
10 eluni 4626 . . . 4 (⟨𝑋, (𝐹𝑋)⟩ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑔(⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}))
119, 10bitri 266 . . 3 (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 ↔ ∃𝑔(⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}))
126, 11sylib 209 . 2 (𝑋 ∈ dom 𝐹 → ∃𝑔(⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}))
13 simprr 780 . . . 4 ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
14 eqid 2802 . . . . 5 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
15 vex 3390 . . . . 5 𝑔 ∈ V
1614, 15wfrlem3a 7646 . . . 4 (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
1713, 16sylib 209 . . 3 ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
18 3simpa 1171 . . . . 5 ((𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))
19 simprlr 789 . . . . . . . . 9 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
20 elssuni 4654 . . . . . . . . . 10 (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝑔 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
2120, 8syl6sseqr 3843 . . . . . . . . 9 (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝑔𝐹)
2219, 21syl 17 . . . . . . . 8 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑔𝐹)
23 predeq3 5891 . . . . . . . . . . 11 (𝑤 = 𝑋 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑋))
2423sseq1d 3823 . . . . . . . . . 10 (𝑤 = 𝑋 → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧))
25 simprrr 791 . . . . . . . . . . 11 (((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))) → ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
2625adantl 469 . . . . . . . . . 10 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
27 simprll 788 . . . . . . . . . . . . 13 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔)
28 df-br 4838 . . . . . . . . . . . . 13 (𝑋𝑔(𝐹𝑋) ↔ ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔)
2927, 28sylibr 225 . . . . . . . . . . . 12 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑋𝑔(𝐹𝑋))
30 fvex 6415 . . . . . . . . . . . . 13 (𝐹𝑋) ∈ V
31 breldmg 5525 . . . . . . . . . . . . 13 ((𝑋 ∈ dom 𝐹 ∧ (𝐹𝑋) ∈ V ∧ 𝑋𝑔(𝐹𝑋)) → 𝑋 ∈ dom 𝑔)
3230, 31mp3an2 1566 . . . . . . . . . . . 12 ((𝑋 ∈ dom 𝐹𝑋𝑔(𝐹𝑋)) → 𝑋 ∈ dom 𝑔)
3329, 32syldan 581 . . . . . . . . . . 11 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑋 ∈ dom 𝑔)
34 simprrl 790 . . . . . . . . . . . 12 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑔 Fn 𝑧)
35 fndm 6195 . . . . . . . . . . . 12 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
3634, 35syl 17 . . . . . . . . . . 11 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → dom 𝑔 = 𝑧)
3733, 36eleqtrd 2883 . . . . . . . . . 10 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑋𝑧)
3824, 26, 37rspcdva 3504 . . . . . . . . 9 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧)
3938, 36sseqtr4d 3833 . . . . . . . 8 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔)
40 fun2ssres 6139 . . . . . . . . 9 ((Fun 𝐹𝑔𝐹 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑋)))
414, 40mp3an1 1565 . . . . . . . 8 ((𝑔𝐹 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑋)))
4222, 39, 41syl2anc 575 . . . . . . 7 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑋)))
4315resex 5642 . . . . . . 7 (𝑔 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V
4442, 43syl6eqel 2889 . . . . . 6 ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
4544expr 446 . . . . 5 ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → ((𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V))
4618, 45syl5 34 . . . 4 ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → ((𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V))
4746exlimdv 2023 . . 3 ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V))
4817, 47mpd 15 . 2 ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹𝑋)⟩ ∈ 𝑔𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
4912, 48exlimddv 2025 1 (𝑋 ∈ dom 𝐹 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2155  {cab 2788  wral 3092  Vcvv 3387  wss 3763  cop 4370   cuni 4623   class class class wbr 4837   Se wse 5262   We wwe 5263  dom cdm 5305  cres 5307  Predcpred 5886  Fun wfun 6089   Fn wfn 6090  cfv 6095  wrecscwrecs 7635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-po 5226  df-so 5227  df-fr 5264  df-se 5265  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-iota 6058  df-fun 6097  df-fn 6098  df-fv 6103  df-wrecs 7636
This theorem is referenced by: (None)
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