Theorem xpf1o 6810

Description: Construct a bijection on a Cartesian product given bijections on the factors. (Contributed by Mario Carneiro, 30-May-2015.)

Hypotheses

Ref	Expression
xpf1o.1	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑋):𝐴–1-1-onto→𝐵)
xpf1o.2	⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑌):𝐶–1-1-onto→𝐷)

Assertion

Ref	Expression
xpf1o	⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦   𝑦,𝑋   𝑥,𝐵   𝑦,𝐷   𝑥,𝑌

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)   𝐷(𝑥)   𝑋(𝑥)   𝑌(𝑦)

Proof of Theorem xpf1o

Dummy variables 𝑡 𝑠 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xp1st 6133	. . . . . 6 ⊢ (𝑢 ∈ (𝐴 × 𝐶) → (1st ‘𝑢) ∈ 𝐴)
2	1	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → (1st ‘𝑢) ∈ 𝐴)
3		xpf1o.1	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑋):𝐴–1-1-onto→𝐵)
4		eqid 2165	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑋) = (𝑥 ∈ 𝐴 ↦ 𝑋)
5	4	f1ompt 5636	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑋):𝐴–1-1-onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋))
6	3, 5	sylib 121	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋))
7	6	simpld 111	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵)
8	7	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵)
9		nfcsb1v 3078	. . . . . . 7 ⊢ Ⅎ𝑥⦋(1st ‘𝑢) / 𝑥⦌𝑋
10	9	nfel1 2319	. . . . . 6 ⊢ Ⅎ𝑥⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵
11		csbeq1a 3054	. . . . . . 7 ⊢ (𝑥 = (1st ‘𝑢) → 𝑋 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋)
12	11	eleq1d 2235	. . . . . 6 ⊢ (𝑥 = (1st ‘𝑢) → (𝑋 ∈ 𝐵 ↔ ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵))
13	10, 12	rspc 2824	. . . . 5 ⊢ ((1st ‘𝑢) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵))
14	2, 8, 13	sylc 62	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵)
15		xp2nd 6134	. . . . . 6 ⊢ (𝑢 ∈ (𝐴 × 𝐶) → (2nd ‘𝑢) ∈ 𝐶)
16	15	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → (2nd ‘𝑢) ∈ 𝐶)
17		xpf1o.2	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑌):𝐶–1-1-onto→𝐷)
18		eqid 2165	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐶 ↦ 𝑌) = (𝑦 ∈ 𝐶 ↦ 𝑌)
19	18	f1ompt 5636	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ↦ 𝑌):𝐶–1-1-onto→𝐷 ↔ (∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷 ∧ ∀𝑤 ∈ 𝐷 ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌))
20	17, 19	sylib 121	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷 ∧ ∀𝑤 ∈ 𝐷 ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌))
21	20	simpld 111	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷)
22	21	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷)
23		nfcsb1v 3078	. . . . . . 7 ⊢ Ⅎ𝑦⦋(2nd ‘𝑢) / 𝑦⦌𝑌
24	23	nfel1 2319	. . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷
25		csbeq1a 3054	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑢) → 𝑌 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌)
26	25	eleq1d 2235	. . . . . 6 ⊢ (𝑦 = (2nd ‘𝑢) → (𝑌 ∈ 𝐷 ↔ ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷))
27	24, 26	rspc 2824	. . . . 5 ⊢ ((2nd ‘𝑢) ∈ 𝐶 → (∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷 → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷))
28	16, 22, 27	sylc 62	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷)
29		opelxpi 4636	. . . 4 ⊢ ((⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵 ∧ ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷) → ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ∈ (𝐵 × 𝐷))
30	14, 28, 29	syl2anc 409	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ∈ (𝐵 × 𝐷))
31	30	ralrimiva 2539	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (𝐴 × 𝐶)⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ∈ (𝐵 × 𝐷))
32	6	simprd 113	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋)
33	32	r19.21bi 2554	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋)
34		reu6 2915	. . . . . . . . 9 ⊢ (∃!𝑥 ∈ 𝐴 𝑧 = 𝑋 ↔ ∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠))
35	33, 34	sylib 121	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠))
36	20	simprd 113	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑤 ∈ 𝐷 ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌)
37	36	r19.21bi 2554	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌)
38		reu6 2915	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ 𝐶 𝑤 = 𝑌 ↔ ∃𝑡 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡))
39	37, 38	sylib 121	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → ∃𝑡 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡))
40	35, 39	anim12dan 590	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)) → (∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ ∃𝑡 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)))
41		reeanv 2635	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐶 (∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)) ↔ (∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ ∃𝑡 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)))
42		pm4.38 595	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)) → ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
43	42	ex 114	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) → ((𝑤 = 𝑌 ↔ 𝑦 = 𝑡) → ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
44	43	ralimdv 2534	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) → (∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡) → ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
45	44	com12 30	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡) → ((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) → ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
46	45	ralimdv 2534	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡) → (∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
47	46	impcom 124	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
48	47	reximi 2563	. . . . . . . . 9 ⊢ (∃𝑡 ∈ 𝐶 (∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)) → ∃𝑡 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
49	48	reximi 2563	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐶 (∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)) → ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
50	41, 49	sylbir 134	. . . . . . 7 ⊢ ((∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ ∃𝑡 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)) → ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
51	40, 50	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)) → ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
52		vex 2729	. . . . . . . . . . . . . . 15 ⊢ 𝑠 ∈ V
53		vex 2729	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
54	52, 53	op1std 6116	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (1st ‘𝑢) = 𝑠)
55	54	csbeq1d 3052	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 = ⦋𝑠 / 𝑥⦌𝑋)
56	55	eqeq2d 2177	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ↔ 𝑧 = ⦋𝑠 / 𝑥⦌𝑋))
57	52, 53	op2ndd 6117	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (2nd ‘𝑢) = 𝑡)
58	57	csbeq1d 3052	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 = ⦋𝑡 / 𝑦⦌𝑌)
59	58	eqeq2d 2177	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ↔ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌))
60	56, 59	anbi12d 465	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → ((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ (𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)))
61		eqeq1 2172	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑢 = 𝑣 ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
62	60, 61	bibi12d 234	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ 𝑢 = 𝑣) ↔ ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)))
63	62	ralxp 4747	. . . . . . . . 9 ⊢ (∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ 𝑢 = 𝑣) ↔ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐶 ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
64		nfv 1516	. . . . . . . . . 10 ⊢ Ⅎ𝑠∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣)
65		nfcv 2308	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐶
66		nfcsb1v 3078	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑠 / 𝑥⦌𝑋
67	66	nfeq2 2320	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑧 = ⦋𝑠 / 𝑥⦌𝑋
68		nfv 1516	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 = ⦋𝑡 / 𝑦⦌𝑌
69	67, 68	nfan 1553	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)
70		nfv 1516	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⟨𝑠, 𝑡⟩ = 𝑣
71	69, 70	nfbi 1577	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)
72	65, 71	nfralxy 2504	. . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑡 ∈ 𝐶 ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)
73		nfv 1516	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣)
74		nfv 1516	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑧 = 𝑋
75		nfcsb1v 3078	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦⦋𝑡 / 𝑦⦌𝑌
76	75	nfeq2 2320	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑤 = ⦋𝑡 / 𝑦⦌𝑌
77	74, 76	nfan 1553	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)
78		nfv 1516	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦⟨𝑥, 𝑡⟩ = 𝑣
79	77, 78	nfbi 1577	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣)
80		csbeq1a 3054	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → 𝑌 = ⦋𝑡 / 𝑦⦌𝑌)
81	80	eqeq2d 2177	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑤 = 𝑌 ↔ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌))
82	81	anbi2d 460	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)))
83		opeq2 3759	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑡⟩)
84	83	eqeq1d 2174	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ ⟨𝑥, 𝑡⟩ = 𝑣))
85	82, 84	bibi12d 234	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣)))
86	73, 79, 85	cbvral 2688	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑡 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣))
87		csbeq1a 3054	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → 𝑋 = ⦋𝑠 / 𝑥⦌𝑋)
88	87	eqeq2d 2177	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑧 = 𝑋 ↔ 𝑧 = ⦋𝑠 / 𝑥⦌𝑋))
89	88	anbi1d 461	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ (𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)))
90		opeq1 3758	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → ⟨𝑥, 𝑡⟩ = ⟨𝑠, 𝑡⟩)
91	90	eqeq1d 2174	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (⟨𝑥, 𝑡⟩ = 𝑣 ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
92	89, 91	bibi12d 234	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣) ↔ ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)))
93	92	ralbidv 2466	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → (∀𝑡 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣) ↔ ∀𝑡 ∈ 𝐶 ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)))
94	86, 93	syl5bb 191	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → (∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑡 ∈ 𝐶 ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)))
95	64, 72, 94	cbvral 2688	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐶 ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
96	63, 95	bitr4i 186	. . . . . . . 8 ⊢ (∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ 𝑢 = 𝑣) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣))
97		eqeq2 2175	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑠, 𝑡⟩))
98		vex 2729	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
99		vex 2729	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
100	98, 99	opth 4215	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑠, 𝑡⟩ ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))
101	97, 100	bitrdi 195	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
102	101	bibi2d 231	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
103	102	2ralbidv 2490	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
104	96, 103	syl5bb 191	. . . . . . 7 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ 𝑢 = 𝑣) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
105	104	rexxp 4748	. . . . . 6 ⊢ (∃𝑣 ∈ (𝐴 × 𝐶)∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ 𝑢 = 𝑣) ↔ ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
106	51, 105	sylibr 133	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)) → ∃𝑣 ∈ (𝐴 × 𝐶)∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ 𝑢 = 𝑣))
107		reu6 2915	. . . . 5 ⊢ (∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ ∃𝑣 ∈ (𝐴 × 𝐶)∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ 𝑢 = 𝑣))
108	106, 107	sylibr 133	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)) → ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
109	108	ralrimivva 2548	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐷 ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
110		eqeq1 2172	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩))
111		vex 2729	. . . . . . 7 ⊢ 𝑧 ∈ V
112		vex 2729	. . . . . . 7 ⊢ 𝑤 ∈ V
113	111, 112	opth 4215	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ↔ (𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
114	110, 113	bitrdi 195	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ↔ (𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌)))
115	114	reubidv 2649	. . . 4 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ↔ ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌)))
116	115	ralxp 4747	. . 3 ⊢ (∀𝑣 ∈ (𝐵 × 𝐷)∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐷 ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
117	109, 116	sylibr 133	. 2 ⊢ (𝜑 → ∀𝑣 ∈ (𝐵 × 𝐷)∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩)
118		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑧⟨𝑋, 𝑌⟩
119		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑤⟨𝑋, 𝑌⟩
120		nfcsb1v 3078	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑋
121		nfcv 2308	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑤 / 𝑦⦌𝑌
122	120, 121	nfop 3774	. . . . 5 ⊢ Ⅎ𝑥⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩
123		nfcv 2308	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌𝑋
124		nfcsb1v 3078	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑌
125	123, 124	nfop 3774	. . . . 5 ⊢ Ⅎ𝑦⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩
126		csbeq1a 3054	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
127		csbeq1a 3054	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝑌 = ⦋𝑤 / 𝑦⦌𝑌)
128		opeq12 3760	. . . . . 6 ⊢ ((𝑋 = ⦋𝑧 / 𝑥⦌𝑋 ∧ 𝑌 = ⦋𝑤 / 𝑦⦌𝑌) → ⟨𝑋, 𝑌⟩ = ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
129	126, 127, 128	syl2an 287	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ⟨𝑋, 𝑌⟩ = ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
130	118, 119, 122, 125, 129	cbvmpo 5921	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐶 ↦ ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
131	111, 112	op1std 6116	. . . . . . 7 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → (1st ‘𝑢) = 𝑧)
132	131	csbeq1d 3052	. . . . . 6 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
133	111, 112	op2ndd 6117	. . . . . . 7 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → (2nd ‘𝑢) = 𝑤)
134	133	csbeq1d 3052	. . . . . 6 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 = ⦋𝑤 / 𝑦⦌𝑌)
135	132, 134	opeq12d 3766	. . . . 5 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ = ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
136	135	mpompt 5934	. . . 4 ⊢ (𝑢 ∈ (𝐴 × 𝐶) ↦ ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐶 ↦ ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
137	130, 136	eqtr4i 2189	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩) = (𝑢 ∈ (𝐴 × 𝐶) ↦ ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩)
138	137	f1ompt 5636	. 2 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷) ↔ (∀𝑢 ∈ (𝐴 × 𝐶)⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ∈ (𝐵 × 𝐷) ∧ ∀𝑣 ∈ (𝐵 × 𝐷)∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩))
139	31, 117, 138	sylanbrc 414	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  ∃!wreu 2446  ⦋csb 3045  ⟨cop 3579   ↦ cmpt 4043   × cxp 4602  –1-1-onto→wf1o 5187  ‘cfv 5188   ∈ cmpo 5844  1^st c1st 6106  2^nd c2nd 6107

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-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

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  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-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109

This theorem is referenced by:  xpen  6811