aciunf1lem - Mathbox for Thierry Arnoux

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Theorem aciunf1lem 33136

Description: Choice in an index union. (Contributed by Thierry Arnoux, 8-Nov-2019.)

**Hypotheses**
Ref	Expression
acunirnmpt.0	⊢ (𝜑 → 𝐴 ∈ 𝑉)
acunirnmpt.1	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)
aciunf1lem.a	⊢ Ⅎ𝑗𝐴
aciunf1lem.1	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)

**Assertion**
Ref	Expression
aciunf1lem	⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘(𝑓‘𝑥)) = 𝑥))

Distinct variable groups: 𝑓,𝑗,𝑥 𝐴,𝑓,𝑥 𝐵,𝑓,𝑥 𝑥,𝑗,𝜑 𝑗,𝑊 Allowed substitution hints: 𝜑(𝑓) 𝐴(𝑗) 𝐵(𝑗) 𝑉(𝑥, 𝑓, 𝑗) 𝑊(𝑥, 𝑓)

Proof of Theorem aciunf1lem Dummy variables 𝑘 𝑦 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		acunirnmpt.0	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		acunirnmpt.1	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)
3		aciunf1lem.a	. . 3 ⊢ Ⅎ𝑗𝐴
4		nfiu1 4986	. . 3 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 𝐵
5		nfcsb1v 3871	. . 3 ⊢ Ⅎ𝑗⦋(𝑔‘𝑥) / 𝑗⦌𝐵
6		eqid 2760	. . 3 ⊢ ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵
7		csbeq1a 3861	. . 3 ⊢ (𝑗 = (𝑔‘𝑥) → 𝐵 = ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
8		aciunf1lem.1	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
9	1, 2, 3, 4, 5, 6, 7, 8	acunirnmpt2f 33135	. 2 ⊢ (𝜑 → ∃𝑔(𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
10		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
11		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴
12		nfra1 3286	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵
13	11, 12	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
14	10, 13	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
15		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝜑
16		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑔
17	16, 4, 3	nff 6699	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴
18		nfcv 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑥
19	18, 5	nfel 2936	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵
20	4, 19	nfralw 3309	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵
21	17, 20	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
22	15, 21	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
23	18, 4	nfel 2936	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵
24	22, 23	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
25		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑗⟨(𝑔‘𝑥), 𝑥⟩
26		nfiu1 4986	. . . . . . . . . 10 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
27	25, 26	nfel 2936	. . . . . . . . 9 ⊢ Ⅎ𝑗⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
28		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
29	28	simpld 500	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → 𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴)
30	29	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴)
31		simpllr 788	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
32	30, 31	ffvelcdmd 7079	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝑔‘𝑥) ∈ 𝐴)
33		fvex 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑔‘𝑥) ∈ V
34	33	snid 4623	. . . . . . . . . . . . . 14 ⊢ (𝑔‘𝑥) ∈ {(𝑔‘𝑥)}
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝑔‘𝑥) ∈ {(𝑔‘𝑥)})
36	28	simprd 501	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
37		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
38		rsp 3250	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵 → (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
39	36, 37, 38	sylc 66	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
40	39	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
41	35, 40	jca 521	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ((𝑔‘𝑥) ∈ {(𝑔‘𝑥)} ∧ 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
42		opelxp 5691	. . . . . . . . . . . 12 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵) ↔ ((𝑔‘𝑥) ∈ {(𝑔‘𝑥)} ∧ 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
43	41, 42	sylibr 237	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
44		sneq 4594	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑔‘𝑥) → {𝑘} = {(𝑔‘𝑥)})
45		csbeq1 3850	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑔‘𝑥) → ⦋𝑘 / 𝑗⦌𝐵 = ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
46	44, 45	xpeq12d 5686	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑔‘𝑥) → ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵) = ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
47	46	eleq2d 2846	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑔‘𝑥) → (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵) ↔ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)))
48	47	rspcev 3576	. . . . . . . . . . 11 ⊢ (((𝑔‘𝑥) ∈ 𝐴 ∧ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → ∃𝑘 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
49	32, 43, 48	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ∃𝑘 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
50		eliun 4955	. . . . . . . . . . 11 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵))
51		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝐴
52		nfv 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵)
53		nfcv 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗{𝑘}
54		nfcsb1v 3871	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐵
55	53, 54	nfxp 5688	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗({𝑘} × ⦋𝑘 / 𝑗⦌𝐵)
56	25, 55	nfel 2936	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵)
57		sneq 4594	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → {𝑗} = {𝑘})
58		csbeq1a 3861	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑗⦌𝐵)
59	57, 58	xpeq12d 5686	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ({𝑗} × 𝐵) = ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
60	59	eleq2d 2846	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵) ↔ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵)))
61	3, 51, 52, 56, 60	cbvrexfw 3303	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
62	50, 61	bitri 278	. . . . . . . . . 10 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
63	49, 62	sylibr 237	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
64		eliun 4955	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↔ ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
65	64	bilani 510	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
66	24, 27, 63, 65	r19.29af2 3270	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
67	66	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
68	14, 67	ralrimi 3260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
69		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
70	33, 69	opth 5452	. . . . . . . . 9 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ ↔ ((𝑔‘𝑥) = (𝑔‘𝑦) ∧ 𝑥 = 𝑦))
71	70	simprbi 503	. . . . . . . 8 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ → 𝑥 = 𝑦)
72	71	rgen2w 3081	. . . . . . 7 ⊢ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵∀𝑦 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ → 𝑥 = 𝑦)
73	72	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵∀𝑦 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ → 𝑥 = 𝑦))
74	68, 73	jca 521	. . . . 5 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → (∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵∀𝑦 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ → 𝑥 = 𝑦)))
75		eqid 2760	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)
76		fveq2 6879	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
77		id 23	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
78	76, 77	opeq12d 4841	. . . . . 6 ⊢ (𝑥 = 𝑦 → ⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩)
79	75, 78	f1mpt 7259	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩):∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ (∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵∀𝑦 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ → 𝑥 = 𝑦)))
80	74, 79	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩):∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
81		opex 5439	. . . . . . . . . 10 ⊢ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ V
82	75	fvmpt2 6999	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ∧ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ V) → ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥) = ⟨(𝑔‘𝑥), 𝑥⟩)
83	81, 82	mpan2 704	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥) = ⟨(𝑔‘𝑥), 𝑥⟩)
84	37, 83	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥) = ⟨(𝑔‘𝑥), 𝑥⟩)
85	84	fveq2d 6883	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → (2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = (2^nd ‘⟨(𝑔‘𝑥), 𝑥⟩))
86	33, 69	op2nd 7996	. . . . . . 7 ⊢ (2^nd ‘⟨(𝑔‘𝑥), 𝑥⟩) = 𝑥
87	85, 86	eqtrdi 2811	. . . . . 6 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → (2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥)
88	87	ex 418	. . . . 5 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → (2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥))
89	14, 88	ralrimi 3260	. . . 4 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥)
90	80, 89	jca 521	. . 3 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩):∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥))
91		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝑘
92	91, 3	nfel 2936	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑘 ∈ 𝐴
93	15, 92	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ 𝐴)
94		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑊
95	54, 94	nfel 2936	. . . . . . . . 9 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊
96	93, 95	nfim 1929	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)
97		eleq1w 2843	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
98	97	anbi2d 642	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝐴) ↔ (𝜑 ∧ 𝑘 ∈ 𝐴)))
99	58	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐵 ∈ 𝑊 ↔ ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊))
100	98, 99	imbi12d 347	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)))
101	96, 100, 8	chvarfv 2276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)
102	101	ralrimiva 3154	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)
103		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑘𝐵
104	3, 51, 103, 54, 58	cbviunf 33030	. . . . . . 7 ⊢ ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ 𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵
105		iunexg 7961	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊) → ∪ 𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ V)
106	104, 105	eqeltrid 2864	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊) → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
107	1, 102, 106	syl2anc 596	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
108		mptexg 7221	. . . . 5 ⊢ (∪ 𝑗 ∈ 𝐴 𝐵 ∈ V → (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) ∈ V)
109		f1eq1 6767	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩):∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
110		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑓
111		nfmpt1 5204	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)
112	110, 111	nfeq 2935	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)
113		fveq1 6878	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → (𝑓‘𝑥) = ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥))
114	113	fveqeq2d 6887	. . . . . . . 8 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → ((2^nd ‘(𝑓‘𝑥)) = 𝑥 ↔ (2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥))
115	112, 114	ralbid 3275	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → (∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘(𝑓‘𝑥)) = 𝑥 ↔ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥))
116	109, 115	anbi12d 644	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘(𝑓‘𝑥)) = 𝑥) ↔ ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩):∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥)))
117	116	spcegv 3551	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) ∈ V → (((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩):∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥) → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘(𝑓‘𝑥)) = 𝑥)))
118	107, 108, 117	3syl 19	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩):∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥) → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘(𝑓‘𝑥)) = 𝑥)))
119	118	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → (((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩):∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥) → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘(𝑓‘𝑥)) = 𝑥)))
120	90, 119	mpd 16	. 2 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘(𝑓‘𝑥)) = 𝑥))
121	9, 120	exlimddv 1968	1 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2^nd ‘(𝑓‘𝑥)) = 𝑥))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 Ⅎwnfc 2907 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⦋csb 3847 ∅c0 4279 {csn 4584 ⟨cop 4590 ∪ ciun 4951 ↦ cmpt 5186 × cxp 5653 ⟶wf 6529 –1-1→wf1 6530 ‘cfv 6533 2^nd c2nd 7986

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-reg 9565 ax-inf2 9621 ax-ac2 10466

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-om 7864 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-en 8954 df-r1 9747 df-rank 9748 df-scott 9869 df-card 9945 df-ac 10120

This theorem is used by: aciunf1 33137

W3C validator