Theorem xpf1o 8678

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 7720	. . . . . 6 ⊢ (𝑢 ∈ (𝐴 × 𝐶) → (1st ‘𝑢) ∈ 𝐴)
2	1	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → (1st ‘𝑢) ∈ 𝐴)
3		xpf1o.1	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑋):𝐴–1-1-onto→𝐵)
4		eqid 2821	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑋) = (𝑥 ∈ 𝐴 ↦ 𝑋)
5	4	f1ompt 6874	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑋):𝐴–1-1-onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋))
6	3, 5	sylib 220	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋))
7	6	simpld 497	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵)
8	7	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵)
9		nfcsb1v 3906	. . . . . . 7 ⊢ Ⅎ𝑥⦋(1st ‘𝑢) / 𝑥⦌𝑋
10	9	nfel1 2994	. . . . . 6 ⊢ Ⅎ𝑥⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵
11		csbeq1a 3896	. . . . . . 7 ⊢ (𝑥 = (1st ‘𝑢) → 𝑋 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋)
12	11	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = (1st ‘𝑢) → (𝑋 ∈ 𝐵 ↔ ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∈ 𝐵))
13	10, 12	rspc 3610	. . . . 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 7721	. . . . . 6 ⊢ (𝑢 ∈ (𝐴 × 𝐶) → (2nd ‘𝑢) ∈ 𝐶)
16	15	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → (2nd ‘𝑢) ∈ 𝐶)
17		xpf1o.2	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑌):𝐶–1-1-onto→𝐷)
18		eqid 2821	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐶 ↦ 𝑌) = (𝑦 ∈ 𝐶 ↦ 𝑌)
19	18	f1ompt 6874	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ↦ 𝑌):𝐶–1-1-onto→𝐷 ↔ (∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷 ∧ ∀𝑤 ∈ 𝐷 ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌))
20	17, 19	sylib 220	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷 ∧ ∀𝑤 ∈ 𝐷 ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌))
21	20	simpld 497	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷)
22	21	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ∀𝑦 ∈ 𝐶 𝑌 ∈ 𝐷)
23		nfcsb1v 3906	. . . . . . 7 ⊢ Ⅎ𝑦⦋(2nd ‘𝑢) / 𝑦⦌𝑌
24	23	nfel1 2994	. . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷
25		csbeq1a 3896	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑢) → 𝑌 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌)
26	25	eleq1d 2897	. . . . . 6 ⊢ (𝑦 = (2nd ‘𝑢) → (𝑌 ∈ 𝐷 ↔ ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 ∈ 𝐷))
27	24, 26	rspc 3610	. . . . 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 5592	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴 × 𝐶)) → ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ∈ (𝐵 × 𝐷))
30	29	ralrimiva 3182	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (𝐴 × 𝐶)⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ∈ (𝐵 × 𝐷))
31	6	simprd 498	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋)
32	31	r19.21bi 3208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 𝑧 = 𝑋)
33		reu6 3716	. . . . . . . . 9 ⊢ (∃!𝑥 ∈ 𝐴 𝑧 = 𝑋 ↔ ∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠))
34	32, 33	sylib 220	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∃𝑠 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑧 = 𝑋 ↔ 𝑥 = 𝑠))
35	20	simprd 498	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑤 ∈ 𝐷 ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌)
36	35	r19.21bi 3208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → ∃!𝑦 ∈ 𝐶 𝑤 = 𝑌)
37		reu6 3716	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ 𝐶 𝑤 = 𝑌 ↔ ∃𝑡 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑤 = 𝑌 ↔ 𝑦 = 𝑡))
38	36, 37	sylib 220	. . . . . . . 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 3367	. . . . . . . 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		pm4.38 636	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) ∧ (𝑤 = 𝑌 ↔ 𝑦 = 𝑡)) → ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
42	41	ex 415	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑋 ↔ 𝑥 = 𝑠) → ((𝑤 = 𝑌 ↔ 𝑦 = 𝑡) → ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
43	42	ralimdv 3178	. . . . . . . . . . . . 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 3178	. . . . . . . . . . 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 410	. . . . . . . . . 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 3243	. . . . . . . . 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 3243	. . . . . . . 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 237	. . . . . . 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 3497	. . . . . . . . . . . . . . 15 ⊢ 𝑠 ∈ V
52		vex 3497	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
53	51, 52	op1std 7698	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (1st ‘𝑢) = 𝑠)
54	53	csbeq1d 3886	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 = ⦋𝑠 / 𝑥⦌𝑋)
55	54	eqeq2d 2832	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ↔ 𝑧 = ⦋𝑠 / 𝑥⦌𝑋))
56	51, 52	op2ndd 7699	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (2nd ‘𝑢) = 𝑡)
57	56	csbeq1d 3886	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 = ⦋𝑡 / 𝑦⦌𝑌)
58	57	eqeq2d 2832	. . . . . . . . . . . 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 2825	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑢 = 𝑣 ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
61	59, 60	bibi12d 348	. . . . . . . . . 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 5711	. . . . . . . . 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 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑠∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣)
64		nfcv 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐶
65		nfcsb1v 3906	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑠 / 𝑥⦌𝑋
66	65	nfeq2 2995	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑧 = ⦋𝑠 / 𝑥⦌𝑋
67		nfv 1911	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 = ⦋𝑡 / 𝑦⦌𝑌
68	66, 67	nfan 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)
69		nfv 1911	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⟨𝑠, 𝑡⟩ = 𝑣
70	68, 69	nfbi 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)
71	64, 70	nfralw 3225	. . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑡 ∈ 𝐶 ((𝑧 = ⦋𝑠 / 𝑥⦌𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)
72		nfv 1911	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣)
73		nfv 1911	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑧 = 𝑋
74		nfcsb1v 3906	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦⦋𝑡 / 𝑦⦌𝑌
75	74	nfeq2 2995	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑤 = ⦋𝑡 / 𝑦⦌𝑌
76	73, 75	nfan 1896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌)
77		nfv 1911	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦⟨𝑥, 𝑡⟩ = 𝑣
78	76, 77	nfbi 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣)
79		csbeq1a 3896	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → 𝑌 = ⦋𝑡 / 𝑦⦌𝑌)
80	79	eqeq2d 2832	. . . . . . . . . . . . . 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 4803	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑡⟩)
83	82	eqeq1d 2823	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ ⟨𝑥, 𝑡⟩ = 𝑣))
84	81, 83	bibi12d 348	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣)))
85	72, 78, 84	cbvralw 3441	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑡 ∈ 𝐶 ((𝑧 = 𝑋 ∧ 𝑤 = ⦋𝑡 / 𝑦⦌𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣))
86		csbeq1a 3896	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → 𝑋 = ⦋𝑠 / 𝑥⦌𝑋)
87	86	eqeq2d 2832	. . . . . . . . . . . . . 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 4802	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → ⟨𝑥, 𝑡⟩ = ⟨𝑠, 𝑡⟩)
90	89	eqeq1d 2823	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (⟨𝑥, 𝑡⟩ = 𝑣 ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
91	88, 90	bibi12d 348	. . . . . . . . . . . 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 3197	. . . . . . . . . . 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	syl5bb 285	. . . . . . . . . 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 3441	. . . . . . . . 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 280	. . . . . . . 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 2833	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑠, 𝑡⟩))
97		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
98		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
99	97, 98	opth 5367	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑠, 𝑡⟩ ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))
100	96, 99	syl6bb 289	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡)))
101	100	bibi2d 345	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑠, 𝑡⟩ → (((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) ↔ (𝑥 = 𝑠 ∧ 𝑦 = 𝑡))))
102	101	2ralbidv 3199	. . . . . . . 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	syl5bb 285	. . . . . . 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 5712	. . . . . 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 236	. . . . 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 3716	. . . . 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 236	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)) → ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
108	107	ralrimivva 3191	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐷 ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
109		eqeq1 2825	. . . . . 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 3497	. . . . . . 7 ⊢ 𝑧 ∈ V
111		vex 3497	. . . . . . 7 ⊢ 𝑤 ∈ V
112	110, 111	opth 5367	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ ↔ (𝑧 = ⦋(1st ‘𝑢) / 𝑥⦌𝑋 ∧ 𝑤 = ⦋(2nd ‘𝑢) / 𝑦⦌𝑌))
113	109, 112	syl6bb 289	. . . . 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 3389	. . . 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 5711	. . 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 236	. 2 ⊢ (𝜑 → ∀𝑣 ∈ (𝐵 × 𝐷)∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩)
117		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑧⟨𝑋, 𝑌⟩
118		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑤⟨𝑋, 𝑌⟩
119		nfcsb1v 3906	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑋
120		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑤 / 𝑦⦌𝑌
121	119, 120	nfop 4818	. . . . 5 ⊢ Ⅎ𝑥⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩
122		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌𝑋
123		nfcsb1v 3906	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑌
124	122, 123	nfop 4818	. . . . 5 ⊢ Ⅎ𝑦⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩
125		csbeq1a 3896	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
126		csbeq1a 3896	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝑌 = ⦋𝑤 / 𝑦⦌𝑌)
127		opeq12 4804	. . . . . 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 7247	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐶 ↦ ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
130	110, 111	op1std 7698	. . . . . . 7 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → (1st ‘𝑢) = 𝑧)
131	130	csbeq1d 3886	. . . . . 6 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⦋(1st ‘𝑢) / 𝑥⦌𝑋 = ⦋𝑧 / 𝑥⦌𝑋)
132	110, 111	op2ndd 7699	. . . . . . 7 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → (2nd ‘𝑢) = 𝑤)
133	132	csbeq1d 3886	. . . . . 6 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⦋(2nd ‘𝑢) / 𝑦⦌𝑌 = ⦋𝑤 / 𝑦⦌𝑌)
134	131, 133	opeq12d 4810	. . . . 5 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩ = ⟨⦋𝑧 / 𝑥⦌𝑋, ⦋𝑤 / 𝑦⦌𝑌⟩)
135	134	mpompt 7265	. . . 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 2847	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩) = (𝑢 ∈ (𝐴 × 𝐶) ↦ ⟨⦋(1st ‘𝑢) / 𝑥⦌𝑋, ⦋(2nd ‘𝑢) / 𝑦⦌𝑌⟩)
137	136	f1ompt 6874	. 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 585	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  ⦋csb 3882  ⟨cop 4572   ↦ cmpt 5145   × cxp 5552  –1-1-onto→wf1o 6353  ‘cfv 6354   ∈ cmpo 7157  1^st c1st 7686  2^nd c2nd 7687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689

This theorem is referenced by:  infxpenc  9443  pwfseqlem5  10084