Theorem tfrcllemsucaccv 6322

Description: Lemma for tfrcl 6332. 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 4380	. . . . 5 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2	1	eleq1d 2235	. . . 4 ⊢ (𝑥 = 𝑧 → (suc 𝑥 ∈ 𝑋 ↔ suc 𝑧 ∈ 𝑋))
3		tfrcllemsucaccv.u	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
4	3	ralrimiva 2539	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ∪ 𝑋 suc 𝑥 ∈ 𝑋)
5		tfrcllemsucaccv.zy	. . . . 5 ⊢ (𝜑 → 𝑧 ∈ 𝑌)
6		tfrcllemsucaccv.yx	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
7		elunii 3794	. . . . 5 ⊢ ((𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋) → 𝑧 ∈ ∪ 𝑋)
8	5, 6, 7	syl2anc 409	. . . 4 ⊢ (𝜑 → 𝑧 ∈ ∪ 𝑋)
9	2, 4, 8	rspcdva 2835	. . 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 304	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋))
16		ordtr1 4366	. . . . 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 6321	. . 3 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆)
21		vex 2729	. . . . . 6 ⊢ 𝑦 ∈ V
22	21	elsuc 4384	. . . . 5 ⊢ (𝑦 ∈ suc 𝑧 ↔ (𝑦 ∈ 𝑧 ∨ 𝑦 = 𝑧))
23		vex 2729	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
24		feq1 5320	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓:𝑥⟶𝑆 ↔ 𝑔:𝑥⟶𝑆))
25		fveq1 5485	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
26		reseq1 4878	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ 𝑦) = (𝑔 ↾ 𝑦))
27	26	fveq2d 5490	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝑔 ↾ 𝑦)))
28	25, 27	eqeq12d 2180	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
29	28	ralbidv 2466	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
30	24, 29	anbi12d 465	. . . . . . . . . . . 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 2467	. . . . . . . . . . 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 2874	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
33	19, 32	sylib 121	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
34		simprrr 530	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))
35		simprrl 529	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → 𝑔:𝑥⟶𝑆)
36		ffn 5337	. . . . . . . . . . . . 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 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → 𝑔:𝑧⟶𝑆)
39		ffn 5337	. . . . . . . . . . . . 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 5289	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝑥 ∧ 𝑔 Fn 𝑧) → 𝑥 = 𝑧)
42	37, 40, 41	syl2anc 409	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))) → 𝑥 = 𝑧)
43	42	raleqdv 2667	. . . . . . . . . 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 146	. . . . . . . . 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 2588	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))
46	45	r19.21bi 2554	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))
47		ordelon 4361	. . . . . . . . . . . . 13 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On)
48	12, 17, 47	syl2anc 409	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑧 ∈ On)
49		onelon 4362	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
50	48, 49	sylan 281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
51		eloni 4353	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → Ord 𝑦)
52		ordirr 4519	. . . . . . . . . . 11 ⊢ (Ord 𝑦 → ¬ 𝑦 ∈ 𝑦)
53	50, 51, 52	3syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ¬ 𝑦 ∈ 𝑦)
54		elequ2 2141	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑦))
55	54	biimpcd 158	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑧 → (𝑧 = 𝑦 → 𝑦 ∈ 𝑦))
56	55	adantl 275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → (𝑧 = 𝑦 → 𝑦 ∈ 𝑦))
57	53, 56	mtod 653	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ¬ 𝑧 = 𝑦)
58	57	neqned 2343	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → 𝑧 ≠ 𝑦)
59		fvunsng 5679	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑧 ≠ 𝑦) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝑔‘𝑦))
60	21, 58, 59	sylancr 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝑔‘𝑦))
61		eloni 4353	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ On → Ord 𝑧)
62	48, 61	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → Ord 𝑧)
63		ordelss 4357	. . . . . . . . . . 11 ⊢ ((Ord 𝑧 ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
64	62, 63	sylan 281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
65		resabs1 4913	. . . . . . . . . 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 4519	. . . . . . . . . . . . 13 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧)
69	62, 68	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧)
70		fsnunres 5687	. . . . . . . . . . . 12 ⊢ ((𝑔 Fn 𝑧 ∧ ¬ 𝑧 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) = 𝑔)
71	67, 69, 70	syl2anc 409	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) = 𝑔)
72	71	reseq1d 4883	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = (𝑔 ↾ 𝑦))
73	72	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧) ↾ 𝑦) = (𝑔 ↾ 𝑦))
74	66, 73	eqtr3d 2200	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦) = (𝑔 ↾ 𝑦))
75	74	fveq2d 5490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)) = (𝐺‘(𝑔 ↾ 𝑦)))
76	46, 60, 75	3eqtr4d 2208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
77		feq2 5321	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑓:𝑥⟶𝑆 ↔ 𝑓:𝑧⟶𝑆))
78	77	imbi1d 230	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
79	78	albidv 1812	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
80	13	3expia 1195	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
81	80	alrimiv 1862	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
82	81	ralrimiva 2539	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
83	79, 82, 17	rspcdva 2835	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
84		feq1 5320	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓:𝑧⟶𝑆 ↔ 𝑔:𝑧⟶𝑆))
85		fveq2 5486	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
86	85	eleq1d 2235	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘𝑔) ∈ 𝑆))
87	84, 86	imbi12d 233	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆)))
88	87	spv 1848	. . . . . . . . . 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 5287	. . . . . . . . . . 11 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
91	67, 90	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑔 = 𝑧)
92	69, 91	neleqtrrd 2265	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
93		fsnunfv 5686	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑌 ∧ (𝐺‘𝑔) ∈ 𝑆 ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑧) = (𝐺‘𝑔))
94	5, 89, 92, 93	syl3anc 1228	. . . . . . . 8 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑧) = (𝐺‘𝑔))
95	94	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑧) = (𝐺‘𝑔))
96		simpr 109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
97	96	fveq2d 5490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑧))
98		reseq2 4879	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑧))
99	98, 71	sylan9eqr 2221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦) = 𝑔)
100	99	fveq2d 5490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)) = (𝐺‘𝑔))
101	95, 97, 100	3eqtr4d 2208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
102	76, 101	jaodan 787	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑧 ∨ 𝑦 = 𝑧)) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
103	22, 102	sylan2b 285	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ suc 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
104	103	ralrimiva 2539	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
105		feq2 5321	. . . . . 6 ⊢ (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑤⟶𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆))
106		raleq 2661	. . . . . 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 465	. . . . 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 2830	. . . 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 5321	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑤⟶𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆))
110		raleq 2661	. . . . . 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 465	. . . . 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 2693	. . . 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 121	. . 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 1226	. 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 2729	. . . . . 6 ⊢ 𝑧 ∈ V
116		opexg 4206	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
117	115, 89, 116	sylancr 411	. . . . 5 ⊢ (𝜑 → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
118		snexg 4163	. . . . 5 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
119	117, 118	syl 14	. . . 4 ⊢ (𝜑 → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
120		unexg 4421	. . . 4 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
121	23, 119, 120	sylancr 411	. . 3 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
122		feq1 5320	. . . . . 6 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑓:𝑥⟶𝑆 ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):𝑥⟶𝑆))
123		fveq1 5485	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑓‘𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})‘𝑦))
124		reseq1 4878	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑓 ↾ 𝑦) = ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦))
125	124	fveq2d 5490	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ↾ 𝑦)))
126	123, 125	eqeq12d 2180	. . . . . . 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 2466	. . . . . 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 465	. . . . 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 2467	. . . 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 2873	. . 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 166	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 968  ∀wal 1341   = wceq 1343   ∈ wcel 2136  {cab 2151   ≠ wne 2336  ∀wral 2444  ∃wrex 2445  Vcvv 2726   ∪ cun 3114   ⊆ wss 3116  {csn 3576  ⟨cop 3579  ∪ cuni 3789  Ord word 4340  Oncon0 4341  suc csuc 4343  dom cdm 4604   ↾ cres 4606  Fun wfun 5182   Fn wfn 5183  ⟶wf 5184  ‘cfv 5188  recscrecs 6272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  tfrcllembacc  6323  tfrcllemres  6330