aciunf1lem - Mathbox for Thierry Arnoux

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Theorem aciunf1lem 30407

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 4953	. . 3 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 𝐵
5		nfcsb1v 3907	. . 3 ⊢ Ⅎ𝑗⦋(𝑔‘𝑥) / 𝑗⦌𝐵
6		eqid 2821	. . 3 ⊢ ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵
7		csbeq1a 3897	. . 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 30406	. 2 ⊢ (𝜑 → ∃𝑔(𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
10		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
11		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴
12		nfra1 3219	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵
13	11, 12	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
14	10, 13	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
15		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝜑
16		nfcv 2977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑔
17	16, 4, 3	nff 6510	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴
18		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑥
19	18, 5	nfel 2992	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵
20	4, 19	nfralw 3225	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵
21	17, 20	nfan 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
22	15, 21	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
23	18, 4	nfel 2992	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵
24	22, 23	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
25		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑗⟨(𝑔‘𝑥), 𝑥⟩
26		nfiu1 4953	. . . . . . . . . 10 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
27	25, 26	nfel 2992	. . . . . . . . 9 ⊢ Ⅎ𝑗⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
28		simplr 767	. . . . . . . . . . . . . 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 497	. . . . . . . . . . . . 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 724	. . . . . . . . . . . 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 774	. . . . . . . . . . . 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	ffvelrnd 6852	. . . . . . . . . . 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 6683	. . . . . . . . . . . . . . 15 ⊢ (𝑔‘𝑥) ∈ V
34	33	snid 4601	. . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . 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 3205	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵 → (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
39	36, 37, 38	sylc 65	. . . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 514	. . . . . . . . . . . 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 5591	. . . . . . . . . . . 12 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵) ↔ ((𝑔‘𝑥) ∈ {(𝑔‘𝑥)} ∧ 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
43	41, 42	sylibr 236	. . . . . . . . . . 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 4577	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑔‘𝑥) → {𝑘} = {(𝑔‘𝑥)})
45		csbeq1 3886	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑔‘𝑥) → ⦋𝑘 / 𝑗⦌𝐵 = ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
46	44, 45	xpeq12d 5586	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑔‘𝑥) → ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵) = ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
47	46	eleq2d 2898	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑔‘𝑥) → (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵) ↔ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)))
48	47	rspcev 3623	. . . . . . . . . . 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 586	. . . . . . . . . 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 4923	. . . . . . . . . . 11 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵))
51		nfcv 2977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝐴
52		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵)
53		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗{𝑘}
54		nfcsb1v 3907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐵
55	53, 54	nfxp 5588	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗({𝑘} × ⦋𝑘 / 𝑗⦌𝐵)
56	25, 55	nfel 2992	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵)
57		sneq 4577	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → {𝑗} = {𝑘})
58		csbeq1a 3897	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑗⦌𝐵)
59	57, 58	xpeq12d 5586	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ({𝑗} × 𝐵) = ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
60	59	eleq2d 2898	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵) ↔ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵)))
61	3, 51, 52, 56, 60	cbvrexfw 3438	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
62	50, 61	bitri 277	. . . . . . . . . 10 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
63	49, 62	sylibr 236	. . . . . . . . 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 4923	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↔ ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
65	64	biimpi 218	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
66	65	adantl 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
67	24, 27, 63, 66	r19.29af2 3330	. . . . . . . 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))
68	67	ex 415	. . . . . . 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)))
69	14, 68	ralrimi 3216	. . . . . 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))
70		vex 3497	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
71	33, 70	opth 5368	. . . . . . . . 9 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ ↔ ((𝑔‘𝑥) = (𝑔‘𝑦) ∧ 𝑥 = 𝑦))
72	71	simprbi 499	. . . . . . . 8 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ → 𝑥 = 𝑦)
73	72	rgen2w 3151	. . . . . . 7 ⊢ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵∀𝑦 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ → 𝑥 = 𝑦)
74	73	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))
75	69, 74	jca 514	. . . . 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)))
76		eqid 2821	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)
77		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
78		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
79	77, 78	opeq12d 4811	. . . . . 6 ⊢ (𝑥 = 𝑦 → ⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩)
80	76, 79	f1mpt 7019	. . . . 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)))
81	75, 80	sylibr 236	. . . 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))
82		opex 5356	. . . . . . . . . 10 ⊢ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ V
83	76	fvmpt2 6779	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ∧ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ V) → ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥) = ⟨(𝑔‘𝑥), 𝑥⟩)
84	82, 83	mpan2 689	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥) = ⟨(𝑔‘𝑥), 𝑥⟩)
85	37, 84	syl 17	. . . . . . . 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⟩)
86	85	fveq2d 6674	. . . . . . 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⟩))
87	33, 70	op2nd 7698	. . . . . . 7 ⊢ (2^nd ‘⟨(𝑔‘𝑥), 𝑥⟩) = 𝑥
88	86, 87	syl6eq 2872	. . . . . 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)
89	88	ex 415	. . . . 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))
90	14, 89	ralrimi 3216	. . . 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)
91	81, 90	jca 514	. . 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))
92		nfcv 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝑘
93	92, 3	nfel 2992	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑘 ∈ 𝐴
94	15, 93	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ 𝐴)
95		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑊
96	54, 95	nfel 2992	. . . . . . . . 9 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊
97	94, 96	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)
98		eleq1w 2895	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
99	98	anbi2d 630	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝐴) ↔ (𝜑 ∧ 𝑘 ∈ 𝐴)))
100	58	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐵 ∈ 𝑊 ↔ ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊))
101	99, 100	imbi12d 347	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)))
102	97, 101, 8	chvarfv 2242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)
103	102	ralrimiva 3182	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)
104		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑘𝐵
105	3, 51, 104, 54, 58	cbviunf 30307	. . . . . . 7 ⊢ ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ 𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵
106		iunexg 7664	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊) → ∪ 𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ V)
107	105, 106	eqeltrid 2917	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊) → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
108	1, 103, 107	syl2anc 586	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
109		mptexg 6984	. . . . 5 ⊢ (∪ 𝑗 ∈ 𝐴 𝐵 ∈ V → (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) ∈ V)
110		f1eq1 6570	. . . . . . 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)))
111		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑓
112		nfmpt1 5164	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)
113	111, 112	nfeq 2991	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)
114		fveq1 6669	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → (𝑓‘𝑥) = ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥))
115	114	fveqeq2d 6678	. . . . . . . 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))
116	113, 115	ralbid 3231	. . . . . . 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))
117	110, 116	anbi12d 632	. . . . . 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)))
118	117	spcegv 3597	. . . . 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)))
119	108, 109, 118	3syl 18	. . . 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)))
120	119	adantr 483	. . 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)))
121	91, 120	mpd 15	. 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))
122	9, 121	exlimddv 1936	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

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Ⅎwnfc 2961 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⦋csb 3883 ∅c0 4291 {csn 4567 ⟨cop 4573 ∪ ciun 4919 ↦ cmpt 5146 × cxp 5553 ⟶wf 6351 –1-1→wf1 6352 ‘cfv 6355 2^nd c2nd 7688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-reg 9056 ax-inf2 9104 ax-ac2 9885

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-en 8510 df-r1 9193 df-rank 9194 df-card 9368 df-ac 9542

This theorem is referenced by: aciunf1 30408

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