Theorem xpf1o 9139

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 8007	. . . . . 6 ⊢ (𝑢 ∈ (𝐴 × 𝐶) → (1st ‘𝑢) ∈ 𝐴)
2	1	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → (1st ‘𝑢) ∈ 𝐴)
3		xpf1o.1	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑋):𝐴–1-1-onto→𝐵)
4		eqid 2733	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑋) = (𝑥 ∈ 𝐴 ↦ 𝑋)
5	4	f1ompt 7111	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑋):𝐴–1-1-onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋))
6	3, 5	sylib 217	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋))
7	6	simpld 496	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵)
8	7	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵)
9		nfcsb1v 3919	. . . . . . 7 ⊢ Ⅎ𝑥⦋(1st ‘𝑢) / 𝑥⦌𝑋
10	9	nfel1 2920	. . . . . 6 ⊢ Ⅎ𝑥⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵
11		csbeq1a 3908	. . . . . . 7 ⊢ (𝑥 = (1st ‘𝑢) → 𝑋 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋)
12	11	eleq1d 2819	. . . . . 6 ⊢ (𝑥 = (1st ‘𝑢) → (𝑋 ∈ 𝐵 ↔ ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵))
13	10, 12	rspc 3601	. . . . 5 ⊢ ((1st ‘𝑢) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵))
14	2, 8, 13	sylc 65	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵)
15		xp2nd 8008	. . . . . 6 ⊢ (𝑢 ∈ (𝐴 × 𝐶) → (2nd ‘𝑢) ∈ 𝐶)
16	15	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → (2nd ‘𝑢) ∈ 𝐶)
17		xpf1o.2	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑌):𝐶–1-1-onto→𝐷)
18		eqid 2733	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐶 ↦ 𝑌) = (𝑦 ∈ 𝐶 ↦ 𝑌)
19	18	f1ompt 7111	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ↦ 𝑌):𝐶–1-1-onto→𝐷 ↔ (∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷 ∧ ∀𝑤 ∈ 𝐷 ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌))
20	17, 19	sylib 217	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷 ∧ ∀𝑤 ∈ 𝐷 ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌))
21	20	simpld 496	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷)
22	21	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷)
23		nfcsb1v 3919	. . . . . . 7 ⊢ Ⅎ𝑦⦋(2nd ‘𝑢) / 𝑦⦌𝑌
24	23	nfel1 2920	. . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷
25		csbeq1a 3908	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑢) → 𝑌 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌)
26	25	eleq1d 2819	. . . . . 6 ⊢ (𝑦 = (2nd ‘𝑢) → (𝑌 ∈ 𝐷 ↔ ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷))
27	24, 26	rspc 3601	. . . . 5 ⊢ ((2nd ‘𝑢) ∈ 𝐶 → (∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷 → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷))
28	16, 22, 27	sylc 65	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷)
29	14, 28	opelxpd 5716	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ∈ (𝐵 × 𝐷))
30	29	ralrimiva 3147	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (𝐴 × 𝐶)⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ∈ (𝐵 × 𝐷))
31	6	simprd 497	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋)
32	31	r19.21bi 3249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋)
33		reu6 3723	. . . . . . . . 9 ⊢ (∃!𝑥 ∈ 𝐴 𝑧 = 𝑋 ↔ ∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠))
34	32, 33	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠))
35	20	simprd 497	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑤 ∈ 𝐷 ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌)
36	35	r19.21bi 3249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌)
37		reu6 3723	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ 𝐶 𝑤 = 𝑌 ↔ ∃𝑡 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡))
38	36, 37	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → ∃𝑡 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡))
39	34, 38	anim12dan 620	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)) → (∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ ∃𝑡 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)))
40		reeanv 3227	. . . . . . . 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)))
41		pm4.38 637	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)) → ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
42	41	ex 414	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) → ((𝑤 = 𝑌 ↔ 𝑦 = 𝑡) → ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
43	42	ralimdv 3170	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) → (∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡) → ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
44	43	com12 32	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡) → ((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) → ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
45	44	ralimdv 3170	. . . . . . . . . . 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))))
46	45	impcom 409	. . . . . . . . . 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)))
47	46	reximi 3085	. . . . . . . . 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)))
48	47	reximi 3085	. . . . . . . 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)))
49	40, 48	sylbir 234	. . . . . . 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)))
50	39, 49	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)) → ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
51		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑠 ∈ V
52		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
53	51, 52	op1std 7985	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (1st ‘𝑢) = 𝑠)
54	53	csbeq1d 3898	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 = ⦋𝑠 / 𝑥⦌𝑋)
55	54	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ↔ 𝑧 = ⦋𝑠 / 𝑥⦌𝑋))
56	51, 52	op2ndd 7986	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (2nd ‘𝑢) = 𝑡)
57	56	csbeq1d 3898	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 = ⦋𝑡 / 𝑦⦌𝑌)
58	57	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ↔ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌))
59	55, 58	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → ((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ (𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)))
60		eqeq1 2737	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑢 = 𝑣 ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
61	59, 60	bibi12d 346	. . . . . . . . . 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)))
62	61	ralxp 5842	. . . . . . . . 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))
63		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑠∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣)
64		nfcv 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐶
65		nfcsb1v 3919	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑠 / 𝑥⦌𝑋
66	65	nfeq2 2921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑧 = ⦋𝑠 / 𝑥⦌𝑋
67		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 = ⦋𝑡 / 𝑦⦌𝑌
68	66, 67	nfan 1903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)
69		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⟨𝑠, 𝑡⟩ = 𝑣
70	68, 69	nfbi 1907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)
71	64, 70	nfralw 3309	. . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑡 ∈ 𝐶 ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)
72		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣)
73		nfv 1918	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑧 = 𝑋
74		nfcsb1v 3919	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦⦋𝑡 / 𝑦⦌𝑌
75	74	nfeq2 2921	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑤 = ⦋𝑡 / 𝑦⦌𝑌
76	73, 75	nfan 1903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)
77		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦⟨𝑥, 𝑡⟩ = 𝑣
78	76, 77	nfbi 1907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣)
79		csbeq1a 3908	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → 𝑌 = ⦋𝑡 / 𝑦⦌𝑌)
80	79	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑤 = 𝑌 ↔ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌))
81	80	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)))
82		opeq2 4875	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑡⟩)
83	82	eqeq1d 2735	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ ⟨𝑥, 𝑡⟩ = 𝑣))
84	81, 83	bibi12d 346	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣)))
85	72, 78, 84	cbvralw 3304	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑡 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣))
86		csbeq1a 3908	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → 𝑋 = ⦋𝑠 / 𝑥⦌𝑋)
87	86	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑧 = 𝑋 ↔ 𝑧 = ⦋𝑠 / 𝑥⦌𝑋))
88	87	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ (𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)))
89		opeq1 4874	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → ⟨𝑥, 𝑡⟩ = ⟨𝑠, 𝑡⟩)
90	89	eqeq1d 2735	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (⟨𝑥, 𝑡⟩ = 𝑣 ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
91	88, 90	bibi12d 346	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣) ↔ ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)))
92	91	ralbidv 3178	. . . . . . . . . . 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)))
93	85, 92	bitrid 283	. . . . . . . . . 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)))
94	63, 71, 93	cbvralw 3304	. . . . . . . . 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))
95	62, 94	bitr4i 278	. . . . . . . 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))
96		eqeq2 2745	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑠, 𝑡⟩))
97		vex 3479	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
98		vex 3479	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
99	97, 98	opth 5477	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑠, 𝑡⟩ ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))
100	96, 99	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
101	100	bibi2d 343	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
102	101	2ralbidv 3219	. . . . . . . 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))))
103	95, 102	bitrid 283	. . . . . . 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))))
104	103	rexxp 5843	. . . . . 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)))
105	50, 104	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)) → ∃𝑣 ∈ (𝐴 × 𝐶)∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌) ↔ 𝑢 = 𝑣))
106		reu6 3723	. . . . 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))
107	105, 106	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)) → ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
108	107	ralrimivva 3201	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐷 ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
109		eqeq1 2737	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩))
110		vex 3479	. . . . . . 7 ⊢ 𝑧 ∈ V
111		vex 3479	. . . . . . 7 ⊢ 𝑤 ∈ V
112	110, 111	opth 5477	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ↔ (𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
113	109, 112	bitrdi 287	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ↔ (𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌)))
114	113	reubidv 3395	. . . 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)))
115	114	ralxp 5842	. . 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))
116	108, 115	sylibr 233	. 2 ⊢ (𝜑 → ∀𝑣 ∈ (𝐵 × 𝐷)∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩)
117		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑧⟨𝑋, 𝑌⟩
118		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑤⟨𝑋, 𝑌⟩
119		nfcsb1v 3919	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑋
120		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑤 / 𝑦⦌𝑌
121	119, 120	nfop 4890	. . . . 5 ⊢ Ⅎ𝑥⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩
122		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌𝑋
123		nfcsb1v 3919	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑌
124	122, 123	nfop 4890	. . . . 5 ⊢ Ⅎ𝑦⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩
125		csbeq1a 3908	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
126		csbeq1a 3908	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝑌 = ⦋𝑤 / 𝑦⦌𝑌)
127		opeq12 4876	. . . . . 6 ⊢ ((𝑋 = ⦋𝑧 / 𝑥⦌𝑋 ∧ 𝑌 = ⦋𝑤 / 𝑦⦌𝑌) → ⟨𝑋, 𝑌⟩ = ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
128	125, 126, 127	syl2an 597	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ⟨𝑋, 𝑌⟩ = ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
129	117, 118, 121, 124, 128	cbvmpo 7503	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐶 ↦ ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
130	110, 111	op1std 7985	. . . . . . 7 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → (1st ‘𝑢) = 𝑧)
131	130	csbeq1d 3898	. . . . . 6 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
132	110, 111	op2ndd 7986	. . . . . . 7 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → (2nd ‘𝑢) = 𝑤)
133	132	csbeq1d 3898	. . . . . 6 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 = ⦋𝑤 / 𝑦⦌𝑌)
134	131, 133	opeq12d 4882	. . . . 5 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ = ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
135	134	mpompt 7522	. . . 4 ⊢ (𝑢 ∈ (𝐴 × 𝐶) ↦ ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐶 ↦ ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
136	129, 135	eqtr4i 2764	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩) = (𝑢 ∈ (𝐴 × 𝐶) ↦ ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩)
137	136	f1ompt 7111	. 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⟩))
138	30, 116, 137	sylanbrc 584	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  ∃!wreu 3375  ⦋csb 3894  ⟨cop 4635   ↦ cmpt 5232   × cxp 5675  –1-1-onto→wf1o 6543  ‘cfv 6544   ∈ cmpo 7411  1^st c1st 7973  2^nd c2nd 7974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976

This theorem is referenced by:  infxpenc  10013  pwfseqlem5  10658