Theorem tfrcllemsucaccv 6625

Description: Lemma for tfrcl 6635. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.)

Hypotheses

Ref	Expression
tfrcl.f	⊢ 𝐹 = recs(𝐺)
tfrcl.g	⊢ (𝜑 → Fun 𝐺)
tfrcl.x	⊢ (𝜑 → Ord 𝑋)
tfrcl.ex	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
tfrcllemsucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfrcllemsucaccv.yx	⊢ (𝜑 → 𝑌 ∈ 𝑋)
tfrcllemsucaccv.zy	⊢ (𝜑 → 𝑧 ∈ 𝑌)
tfrcllemsucaccv.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfrcllemsucaccv.gfn	⊢ (𝜑 → 𝑔:𝑧⟶𝑆)
tfrcllemsucaccv.gacc	⊢ (𝜑 → 𝑔 ∈ 𝐴)

Assertion

Ref	Expression
tfrcllemsucaccv	⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴)

Distinct variable groups:   𝑓,𝐺,𝑥,𝑦   𝑆,𝑓,𝑥   𝑓,𝑋,𝑥   𝑓,𝑔,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦   𝑧,𝑓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧, 𝑔)   𝐴(𝑥, 𝑦, 𝑧, 𝑓, 𝑔)   𝑆(𝑦, 𝑧, 𝑔)   𝐹(𝑥, 𝑦, 𝑧, 𝑓, 𝑔)   𝐺(𝑧, 𝑔)   𝑋(𝑦, 𝑧, 𝑔)   𝑌(𝑥, 𝑦, 𝑧, 𝑓, 𝑔)

Proof of Theorem tfrcllemsucaccv

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suceq 4547	. . . . 5 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2	1	eleq1d 2307	. . . 4 ⊢ (𝑥 = 𝑧 → (suc 𝑥 ∈ 𝑋 ↔ suc 𝑧 ∈ 𝑋))
3		tfrcllemsucaccv.u	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
4	3	ralrimiva 2623	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ∪ 𝑋 suc 𝑥 ∈ 𝑋)
5		tfrcllemsucaccv.zy	. . . . 5 ⊢ (𝜑 → 𝑧 ∈ 𝑌)
6		tfrcllemsucaccv.yx	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
7		elunii 3940	. . . . 5 ⊢ ((𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋) → 𝑧 ∈ ∪ 𝑋)
8	5, 6, 7	syl2anc 415	. . . 4 ⊢ (𝜑 → 𝑧 ∈ ∪ 𝑋)
9	2, 4, 8	rspcdva 2934	. . 3 ⊢ (𝜑 → suc 𝑧 ∈ 𝑋)
10		tfrcl.f	. . . 4 ⊢ 𝐹 = recs(𝐺)
11		tfrcl.g	. . . 4 ⊢ (𝜑 → Fun 𝐺)
12		tfrcl.x	. . . 4 ⊢ (𝜑 → Ord 𝑋)
13		tfrcl.ex	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
14		tfrcllemsucfn.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
15	5, 6	jca 306	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋))
16		ordtr1 4533	. . . . 5 ⊢ (Ord 𝑋 → ((𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋) → 𝑧 ∈ 𝑋))
17	12, 15, 16	sylc 62	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝑋)
18		tfrcllemsucaccv.gfn	. . . 4 ⊢ (𝜑 → 𝑔:𝑧⟶𝑆)
19		tfrcllemsucaccv.gacc	. . . 4 ⊢ (𝜑 → 𝑔 ∈ 𝐴)
20	10, 11, 12, 13, 14, 17, 18, 19	tfrcllemsucfn 6624	. . 3 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆)
21		vex 2824	. . . . . 6 ⊢ 𝑦 ∈ V
22	21	elsuc 4551	. . . . 5 ⊢ (𝑦 ∈ suc 𝑧 ↔ (𝑦 ∈ 𝑧 ∨ 𝑦 = 𝑧))
23		vex 2824	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
24		feq1 5516	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓:𝑥⟶𝑆 ↔ 𝑔:𝑥⟶𝑆))
25		fveq1 5694	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
26		reseq1 5057	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ 𝑦) = (𝑔 ↾ 𝑦))
27	26	fveq2d 5699	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝑔 ↾ 𝑦)))
28	25, 27	eqeq12d 2253	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
29	28	ralbidv 2550	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
30	24, 29	anbi12d 477	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))))
31	30	rexbidv 2551	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))))
32	23, 31, 14	elab2 2974	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
33	19, 32	sylib 122	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
34		simprrr 546	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))
35		simprrl 545	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → 𝑔:𝑥⟶𝑆)
36		ffn 5533	. . . . . . . . . . . . 13 ⊢ (𝑔:𝑥⟶𝑆 → 𝑔 Fn 𝑥)
37	35, 36	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → 𝑔 Fn 𝑥)
38	18	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → 𝑔:𝑧⟶𝑆)
39		ffn 5533	. . . . . . . . . . . . 13 ⊢ (𝑔:𝑧⟶𝑆 → 𝑔 Fn 𝑧)
40	38, 39	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → 𝑔 Fn 𝑧)
41		fndmu 5484	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝑥 ∧ 𝑔 Fn 𝑧) → 𝑥 = 𝑧)
42	37, 40, 41	syl2anc 415	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → 𝑥 = 𝑧)
43	42	raleqdv 2755	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → (∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
44	34, 43	mpbid 147	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))
45	33, 44	rexlimddv 2673	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))
46	45	r19.21bi 2638	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))
47		ordelon 4528	. . . . . . . . . . . . 13 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On)
48	12, 17, 47	syl2anc 415	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑧 ∈ On)
49		onelon 4529	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
50	48, 49	sylan 283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
51		eloni 4520	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → Ord 𝑦)
52		ordirr 4689	. . . . . . . . . . 11 ⊢ (Ord 𝑦 → ¬ 𝑦 ∈ 𝑦)
53	50, 51, 52	3syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ¬ 𝑦 ∈ 𝑦)
54		elequ2 2214	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑦))
55	54	biimpcd 159	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑧 → (𝑧 = 𝑦 → 𝑦 ∈ 𝑦))
56	55	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → (𝑧 = 𝑦 → 𝑦 ∈ 𝑦))
57	53, 56	mtod 673	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ¬ 𝑧 = 𝑦)
58	57	neqned 2427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → 𝑧 ≠ 𝑦)
59		fvunsng 5909	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑧 ≠ 𝑦) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝑔‘𝑦))
60	21, 58, 59	sylancr 418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝑔‘𝑦))
61		eloni 4520	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ On → Ord 𝑧)
62	48, 61	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → Ord 𝑧)
63		ordelss 4524	. . . . . . . . . . 11 ⊢ ((Ord 𝑧 ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
64	62, 63	sylan 283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
65		resabs1 5092	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))
66	64, 65	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))
67	18, 39	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑔 Fn 𝑧)
68		ordirr 4689	. . . . . . . . . . . . 13 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧)
69	62, 68	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧)
70		fsnunres 5917	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝑧 ∧ ¬ 𝑧 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) = 𝑔)
71	67, 69, 70	syl2anc 415	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) = 𝑔)
72	71	reseq1d 5062	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = (𝑔 ↾ 𝑦))
73	72	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = (𝑔 ↾ 𝑦))
74	66, 73	eqtr3d 2273	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦) = (𝑔 ↾ 𝑦))
75	74	fveq2d 5699	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)) = (𝐺‘(𝑔 ↾ 𝑦)))
76	46, 60, 75	3eqtr4d 2281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
77		feq2 5517	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑓:𝑥⟶𝑆 ↔ 𝑓:𝑧⟶𝑆))
78	77	imbi1d 231	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
79	78	albidv 1877	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
80	13	3expia 1236	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
81	80	alrimiv 1927	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
82	81	ralrimiva 2623	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
83	79, 82, 17	rspcdva 2934	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
84		feq1 5516	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓:𝑧⟶𝑆 ↔ 𝑔:𝑧⟶𝑆))
85		fveq2 5695	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
86	85	eleq1d 2307	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘𝑔) ∈ 𝑆))
87	84, 86	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆)))
88	87	spv 1913	. . . . . . . . . 10 ⊢ (∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) → (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
89	83, 18, 88	sylc 62	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑔) ∈ 𝑆)
90		fndm 5480	. . . . . . . . . . 11 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
91	67, 90	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑔 = 𝑧)
92	69, 91	neleqtrrd 2337	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
93		fsnunfv 5916	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑌 ∧ (𝐺‘𝑔) ∈ 𝑆 ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑧) = (𝐺‘𝑔))
94	5, 89, 92, 93	syl3anc 1278	. . . . . . . 8 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑧) = (𝐺‘𝑔))
95	94	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑧) = (𝐺‘𝑔))
96		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
97	96	fveq2d 5699	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑧))
98		reseq2 5058	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧))
99	98, 71	sylan9eqr 2293	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦) = 𝑔)
100	99	fveq2d 5699	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)) = (𝐺‘𝑔))
101	95, 97, 100	3eqtr4d 2281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
102	76, 101	jaodan 809	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑧 ∨ 𝑦 = 𝑧)) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
103	22, 102	sylan2b 287	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ suc 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
104	103	ralrimiva 2623	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
105		feq2 5517	. . . . . 6 ⊢ (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑤⟶𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆))
106		raleq 2749	. . . . . 6 ⊢ (𝑤 = suc 𝑧 → (∀𝑦 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)) ↔ ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))))
107	105, 106	anbi12d 477	. . . . 5 ⊢ (𝑤 = suc 𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑤⟶𝑆 ∧ ∀𝑦 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆 ∧ ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))))
108	107	rspcev 2929	. . . 4 ⊢ ((suc 𝑧 ∈ 𝑋 ∧ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆 ∧ ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))) → ∃𝑤 ∈ 𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑤⟶𝑆 ∧ ∀𝑦 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))))
109		feq2 5517	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑤⟶𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆))
110		raleq 2749	. . . . . 6 ⊢ (𝑤 = 𝑥 → (∀𝑦 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))))
111	109, 110	anbi12d 477	. . . . 5 ⊢ (𝑤 = 𝑥 → (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑤⟶𝑆 ∧ ∀𝑦 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))))
112	111	cbvrexv 2787	. . . 4 ⊢ (∃𝑤 ∈ 𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑤⟶𝑆 ∧ ∀𝑦 ∈ 𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))) ↔ ∃𝑥 ∈ 𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))))
113	108, 112	sylib 122	. . 3 ⊢ ((suc 𝑧 ∈ 𝑋 ∧ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆 ∧ ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))) → ∃𝑥 ∈ 𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))))
114	9, 20, 104, 113	syl12anc 1276	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))))
115		vex 2824	. . . . . 6 ⊢ 𝑧 ∈ V
116		opexg 4368	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
117	115, 89, 116	sylancr 418	. . . . 5 ⊢ (𝜑 → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
118		snexg 4321	. . . . 5 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
119	117, 118	syl 14	. . . 4 ⊢ (𝜑 → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
120		unexg 4589	. . . 4 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
121	23, 119, 120	sylancr 418	. . 3 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
122		feq1 5516	. . . . . 6 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑓:𝑥⟶𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆))
123		fveq1 5694	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑓‘𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦))
124		reseq1 5057	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑓 ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))
125	124	fveq2d 5699	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
126	123, 125	eqeq12d 2253	. . . . . . 7 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))))
127	126	ralbidv 2550	. . . . . 6 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))))
128	122, 127	anbi12d 477	. . . . 5 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → ((𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))))
129	128	rexbidv 2551	. . . 4 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥 ∈ 𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))))
130	129, 14	elab2g 2973	. . 3 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))))
131	121, 130	syl 14	. 2 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))))
132	114, 131	mpbird 167	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∧ w3a 1009  ∀wal 1400   = wceq 1402   ∈ wcel 2209  {cab 2224   ≠ wne 2420  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  {csn 3709  ⟨cop 3712  ∪ cuni 3935  Ord word 4507  Oncon0 4508  suc csuc 4510  dom cdm 4774   ↾ cres 4776  Fun wfun 5371   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  recscrecs 6575

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385

This theorem is used by:  tfrcllembacc  6626  tfrcllemres  6633