wfrlem17 - Metamath Proof Explorer

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Theorem wfrlem17 8049

Description: Without using ax-rep 5164, 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.

Step	Hyp	Ref	Expression
1		wfrlem17.1	. . . . 5 ⊢ 𝑅 We 𝐴
2		wfrlem17.2	. . . . 5 ⊢ 𝑅 Se 𝐴
3		wfrlem17.3	. . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4	1, 2, 3	wfrfun 8043	. . . 4 ⊢ Fun 𝐹
5		funfvop 6848	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹)
6	4, 5	mpan 690	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹)
7		df-wrecs 8025	. . . . . 6 ⊢ wrecs(𝑅, 𝐴, 𝐺) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
8	3, 7	eqtri 2759	. . . . 5 ⊢ 𝐹 = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
9	8	eleq2i 2822	. . . 4 ⊢ (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
10		eluni 4808	. . . 4 ⊢ (⟨𝑋, (𝐹‘𝑋)⟩ ∈ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑔(⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}))
11	9, 10	bitri 278	. . 3 ⊢ (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹 ↔ ∃𝑔(⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}))
12	6, 11	sylib 221	. 2 ⊢ (𝑋 ∈ dom 𝐹 → ∃𝑔(⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}))
13		simprr 773	. . . 4 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
14		eqid 2736	. . . . 5 ⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
15		vex 3402	. . . . 5 ⊢ 𝑔 ∈ V
16	14, 15	wfrlem3a 8035	. . . 4 ⊢ (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
17	13, 16	sylib 221	. . 3 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
18		3simpa 1150	. . . . 5 ⊢ ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))
19		simprlr 780	. . . . . . . . 9 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
20		elssuni 4837	. . . . . . . . . 10 ⊢ (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝑔 ⊆ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
21	20, 8	sseqtrrdi 3938	. . . . . . . . 9 ⊢ (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝑔 ⊆ 𝐹)
22	19, 21	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑔 ⊆ 𝐹)
23		predeq3 6144	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑋))
24	23	sseq1d 3918	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧))
25		simprrr 782	. . . . . . . . . . 11 ⊢ (((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))) → ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
26	25	adantl 485	. . . . . . . . . 10 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
27		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔)
28		df-br 5040	. . . . . . . . . . . . 13 ⊢ (𝑋𝑔(𝐹‘𝑋) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔)
29	27, 28	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑋𝑔(𝐹‘𝑋))
30		fvex 6708	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑋) ∈ V
31		breldmg 5763	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝑔(𝐹‘𝑋)) → 𝑋 ∈ dom 𝑔)
32	30, 31	mp3an2 1451	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ dom 𝐹 ∧ 𝑋𝑔(𝐹‘𝑋)) → 𝑋 ∈ dom 𝑔)
33	29, 32	syldan 594	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑋 ∈ dom 𝑔)
34		simprrl 781	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑔 Fn 𝑧)
35	34	fndmd 6461	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → dom 𝑔 = 𝑧)
36	33, 35	eleqtrd 2833	. . . . . . . . . 10 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → 𝑋 ∈ 𝑧)
37	24, 26, 36	rspcdva 3529	. . . . . . . . 9 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧)
38	37, 35	sseqtrrd 3928	. . . . . . . 8 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔)
39		fun2ssres 6403	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑔 ⊆ 𝐹 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑋)))
40	4, 22, 38, 39	mp3an2i 1468	. . . . . . 7 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑋)))
41	15	resex 5884	. . . . . . 7 ⊢ (𝑔 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V
42	40, 41	eqeltrdi 2839	. . . . . 6 ⊢ ((𝑋 ∈ dom 𝐹 ∧ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}) ∧ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
43	42	expr 460	. . . . 5 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V))
44	18, 43	syl5 34	. . . 4 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V))
45	44	exlimdv 1941	. . 3 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V))
46	17, 45	mpd 15	. 2 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝑔 ∧ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
47	12, 46	exlimddv 1943	1 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2112 {cab 2714 ∀wral 3051 Vcvv 3398 ⊆ wss 3853 ⟨cop 4533 ∪ cuni 4805 class class class wbr 5039 Se wse 5492 We wwe 5493 dom cdm 5536 ↾ cres 5538 Predcpred 6139 Fun wfun 6352 Fn wfn 6353 ‘cfv 6358 wrecscwrecs 8024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-iota 6316 df-fun 6360 df-fn 6361 df-fv 6366 df-wrecs 8025

This theorem is referenced by: (None)

W3C validator