Theorem aciunf1lem 31578

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(2nd ‘(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 4988	. . 3 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 𝐵
5		nfcsb1v 3880	. . 3 ⊢ Ⅎ𝑗⦋(𝑔‘𝑥) / 𝑗⦌𝐵
6		eqid 2736	. . 3 ⊢ ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵
7		csbeq1a 3869	. . 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 31577	. 2 ⊢ (𝜑 → ∃𝑔(𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
10		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
11		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴
12		nfra1 3267	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵
13	11, 12	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
14	10, 13	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
15		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝜑
16		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑔
17	16, 4, 3	nff 6664	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴
18		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑥
19	18, 5	nfel 2921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵
20	4, 19	nfralw 3294	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵
21	17, 20	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
22	15, 21	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
23	18, 4	nfel 2921	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵
24	22, 23	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
25		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑗⟨(𝑔‘𝑥), 𝑥⟩
26		nfiu1 4988	. . . . . . . . . 10 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
27	25, 26	nfel 2921	. . . . . . . . 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 495	. . . . . . . . . . . . 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	ffvelcdmd 7036	. . . . . . . . . . 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 6855	. . . . . . . . . . . . . . 15 ⊢ (𝑔‘𝑥) ∈ V
34	33	snid 4622	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . . 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 512	. . . . . . . . . . . 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 5669	. . . . . . . . . . . 12 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵) ↔ ((𝑔‘𝑥) ∈ {(𝑔‘𝑥)} ∧ 𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
43	41, 42	sylibr 233	. . . . . . . . . . 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 4596	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑔‘𝑥) → {𝑘} = {(𝑔‘𝑥)})
45		csbeq1 3858	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑔‘𝑥) → ⦋𝑘 / 𝑗⦌𝐵 = ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)
46	44, 45	xpeq12d 5664	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑔‘𝑥) → ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵) = ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵))
47	46	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑔‘𝑥) → (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵) ↔ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({(𝑔‘𝑥)} × ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)))
48	47	rspcev 3581	. . . . . . . . . . 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 584	. . . . . . . . . 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 4958	. . . . . . . . . . 11 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵))
51		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝐴
52		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵)
53		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗{𝑘}
54		nfcsb1v 3880	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐵
55	53, 54	nfxp 5666	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗({𝑘} × ⦋𝑘 / 𝑗⦌𝐵)
56	25, 55	nfel 2921	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵)
57		sneq 4596	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → {𝑗} = {𝑘})
58		csbeq1a 3869	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑗⦌𝐵)
59	57, 58	xpeq12d 5664	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ({𝑗} × 𝐵) = ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
60	59	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵) ↔ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵)))
61	3, 51, 52, 56, 60	cbvrexfw 3288	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
62	50, 61	bitri 274	. . . . . . . . . 10 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐴 ⟨(𝑔‘𝑥), 𝑥⟩ ∈ ({𝑘} × ⦋𝑘 / 𝑗⦌𝐵))
63	49, 62	sylibr 233	. . . . . . . . 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 4958	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↔ ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
65	64	biimpi 215	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
66	65	adantl 482	. . . . . . . . 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 3250	. . . . . . . 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 413	. . . . . . 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 3240	. . . . . 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 3449	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
71	33, 70	opth 5433	. . . . . . . . 9 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ ↔ ((𝑔‘𝑥) = (𝑔‘𝑦) ∧ 𝑥 = 𝑦))
72	71	simprbi 497	. . . . . . . 8 ⊢ (⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩ → 𝑥 = 𝑦)
73	72	rgen2w 3069	. . . . . . 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 512	. . . . 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 2736	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)
77		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
78		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
79	77, 78	opeq12d 4838	. . . . . 6 ⊢ (𝑥 = 𝑦 → ⟨(𝑔‘𝑥), 𝑥⟩ = ⟨(𝑔‘𝑦), 𝑦⟩)
80	76, 79	f1mpt 7208	. . . . 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 233	. . . 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 5421	. . . . . . . . . 10 ⊢ ⟨(𝑔‘𝑥), 𝑥⟩ ∈ V
83	76	fvmpt2 6959	. . . . . . . . . 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 6846	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → (2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = (2nd ‘⟨(𝑔‘𝑥), 𝑥⟩))
87	33, 70	op2nd 7930	. . . . . . 7 ⊢ (2nd ‘⟨(𝑔‘𝑥), 𝑥⟩) = 𝑥
88	86, 87	eqtrdi 2792	. . . . . 6 ⊢ (((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) ∧ 𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → (2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥)
89	88	ex 413	. . . . 5 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → (2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥))
90	14, 89	ralrimi 3240	. . . 4 ⊢ ((𝜑 ∧ (𝑔:∪ 𝑗 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝑥 ∈ ⦋(𝑔‘𝑥) / 𝑗⦌𝐵)) → ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥)
91	81, 90	jca 512	. . 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(2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥))
92		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝑘
93	92, 3	nfel 2921	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑘 ∈ 𝐴
94	15, 93	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ 𝐴)
95		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑊
96	54, 95	nfel 2921	. . . . . . . . 9 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊
97	94, 96	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)
98		eleq1w 2820	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
99	98	anbi2d 629	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝐴) ↔ (𝜑 ∧ 𝑘 ∈ 𝐴)))
100	58	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐵 ∈ 𝑊 ↔ ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊))
101	99, 100	imbi12d 344	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)))
102	97, 101, 8	chvarfv 2233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)
103	102	ralrimiva 3143	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊)
104		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑘𝐵
105	3, 51, 104, 54, 58	cbviunf 31474	. . . . . . 7 ⊢ ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ 𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵
106		iunexg 7896	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊) → ∪ 𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ V)
107	105, 106	eqeltrid 2842	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 ⦋𝑘 / 𝑗⦌𝐵 ∈ 𝑊) → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
108	1, 103, 107	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
109		mptexg 7171	. . . . 5 ⊢ (∪ 𝑗 ∈ 𝐴 𝐵 ∈ V → (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) ∈ V)
110		f1eq1 6733	. . . . . . 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 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑓
112		nfmpt1 5213	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)
113	111, 112	nfeq 2920	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)
114		fveq1 6841	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → (𝑓‘𝑥) = ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥))
115	114	fveqeq2d 6850	. . . . . . . 8 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → ((2nd ‘(𝑓‘𝑥)) = 𝑥 ↔ (2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥))
116	113, 115	ralbid 3256	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → (∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑥)) = 𝑥 ↔ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥))
117	110, 116	anbi12d 631	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩) → ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑥)) = 𝑥) ↔ ((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩):∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥)))
118	117	spcegv 3556	. . . . 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(2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥) → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(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(2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥) → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑥)) = 𝑥)))
120	119	adantr 481	. . 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(2nd ‘((𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↦ ⟨(𝑔‘𝑥), 𝑥⟩)‘𝑥)) = 𝑥) → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(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(2nd ‘(𝑓‘𝑥)) = 𝑥))
122	9, 121	exlimddv 1938	1 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑥)) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Ⅎwnfc 2887   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  Vcvv 3445  ⦋csb 3855  ∅c0 4282  {csn 4586  ⟨cop 4592  ∪ ciun 4954   ↦ cmpt 5188   × cxp 5631  ⟶wf 6492  –1-1→wf1 6493  ‘cfv 6496  2^nd c2nd 7920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577  ax-ac2 10399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-en 8884  df-r1 9700  df-rank 9701  df-card 9875  df-ac 10052

This theorem is referenced by:  aciunf1  31579