Theorem f1od2 6140

Description: Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)

Hypotheses

Ref	Expression
f1od2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
f1od2.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊)
f1od2.3	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌))
f1od2.4	⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))

Assertion

Ref	Expression
f1od2	⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑧,𝐶   𝑥,𝐷,𝑦,𝑧   𝑥,𝐼,𝑦   𝑥,𝐽,𝑦   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧)   𝐼(𝑧)   𝐽(𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)

Proof of Theorem f1od2

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1od2.2	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊)
2	1	ralrimivva 2517	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑊)
3		f1od2.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	fnmpo 6108	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑊 → 𝐹 Fn (𝐴 × 𝐵))
5	2, 4	syl 14	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))
6		f1od2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌))
7		opelxpi 4579	. . . . . 6 ⊢ ((𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
8	6, 7	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
9	8	ralrimiva 2508	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
10		eqid 2140	. . . . 5 ⊢ (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) = (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩)
11	10	fnmpt 5257	. . . 4 ⊢ (∀𝑧 ∈ 𝐷 ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌) → (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷)
12	9, 11	syl 14	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷)
13		elxp7 6076	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)))
14	13	anbi1i 454	. . . . . . 7 ⊢ ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶) ↔ ((𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶))
15		anass 399	. . . . . . . . 9 ⊢ (((𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑎 ∈ (V × V) ∧ (((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶)))
16		f1od2.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))
17	16	sbcbidv 2971	. . . . . . . . . . . 12 ⊢ (𝜑 → ([(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))
18	17	sbcbidv 2971	. . . . . . . . . . 11 ⊢ (𝜑 → ([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))
19		sbcan 2955	. . . . . . . . . . . . . 14 ⊢ ([(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶))
20		sbcan 2955	. . . . . . . . . . . . . . . 16 ⊢ ([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ ([(2nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵))
21		vex 2692	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑎 ∈ V
22		2ndexg 6074	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ V → (2nd ‘𝑎) ∈ V)
23	21, 22	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘𝑎) ∈ V
24		sbcg 2982	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑎) ∈ V → ([(2nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
25	23, 24	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ([(2nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
26		sbcel1v 2975	. . . . . . . . . . . . . . . . 17 ⊢ ([(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵 ↔ (2nd ‘𝑎) ∈ 𝐵)
27	25, 26	anbi12i 456	. . . . . . . . . . . . . . . 16 ⊢ (([(2nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵))
28	20, 27	bitri 183	. . . . . . . . . . . . . . 15 ⊢ ([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵))
29		sbceq2g 3029	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑎) ∈ V → ([(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶))
30	23, 29	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶)
31	28, 30	anbi12i 456	. . . . . . . . . . . . . 14 ⊢ (([(2nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(2nd ‘𝑎) / 𝑦]𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶))
32	19, 31	bitri 183	. . . . . . . . . . . . 13 ⊢ ([(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶))
33	32	sbcbii 2972	. . . . . . . . . . . 12 ⊢ ([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(1st ‘𝑎) / 𝑥]((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶))
34		sbcan 2955	. . . . . . . . . . . 12 ⊢ ([(1st ‘𝑎) / 𝑥]((𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶) ↔ ([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ [(1st ‘𝑎) / 𝑥]𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶))
35		sbcan 2955	. . . . . . . . . . . . . 14 ⊢ ([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ↔ ([(1st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵))
36		sbcel1v 2975	. . . . . . . . . . . . . . 15 ⊢ ([(1st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ↔ (1st ‘𝑎) ∈ 𝐴)
37		1stexg 6073	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ V → (1st ‘𝑎) ∈ V)
38	21, 37	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘𝑎) ∈ V
39		sbcg 2982	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑎) ∈ V → ([(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵 ↔ (2nd ‘𝑎) ∈ 𝐵))
40	38, 39	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵 ↔ (2nd ‘𝑎) ∈ 𝐵)
41	36, 40	anbi12i 456	. . . . . . . . . . . . . 14 ⊢ (([(1st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) ∈ 𝐵) ↔ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵))
42	35, 41	bitri 183	. . . . . . . . . . . . 13 ⊢ ([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ↔ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵))
43		sbceq2g 3029	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑎) ∈ V → ([(1st ‘𝑎) / 𝑥]𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶))
44	38, 43	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([(1st ‘𝑎) / 𝑥]𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶)
45	42, 44	anbi12i 456	. . . . . . . . . . . 12 ⊢ (([(1st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ [(1st ‘𝑎) / 𝑥]𝑧 = ⦋(2nd ‘𝑎) / 𝑦⦌𝐶) ↔ (((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶))
46	33, 34, 45	3bitri 205	. . . . . . . . . . 11 ⊢ ([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶))
47		sbcan 2955	. . . . . . . . . . . . . 14 ⊢ ([(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ ([(2nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
48		sbcg 2982	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑎) ∈ V → ([(2nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷))
49	23, 48	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(2nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷)
50		sbcan 2955	. . . . . . . . . . . . . . . 16 ⊢ ([(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ ([(2nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽))
51		sbcg 2982	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑎) ∈ V → ([(2nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ↔ 𝑥 = 𝐼))
52	23, 51	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ([(2nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ↔ 𝑥 = 𝐼)
53		sbceq1g 3027	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑎) ∈ V → ([(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ ⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = 𝐽))
54	23, 53	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ([(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ ⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = 𝐽)
55		csbvarg 3035	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘𝑎) ∈ V → ⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = (2nd ‘𝑎))
56	23, 55	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = (2nd ‘𝑎)
57	56	eqeq1i 2148	. . . . . . . . . . . . . . . . . 18 ⊢ (⦋(2nd ‘𝑎) / 𝑦⦌𝑦 = 𝐽 ↔ (2nd ‘𝑎) = 𝐽)
58	54, 57	bitri 183	. . . . . . . . . . . . . . . . 17 ⊢ ([(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ (2nd ‘𝑎) = 𝐽)
59	52, 58	anbi12i 456	. . . . . . . . . . . . . . . 16 ⊢ (([(2nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2nd ‘𝑎) / 𝑦]𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))
60	50, 59	bitri 183	. . . . . . . . . . . . . . 15 ⊢ ([(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))
61	49, 60	anbi12i 456	. . . . . . . . . . . . . 14 ⊢ (([(2nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))
62	47, 61	bitri 183	. . . . . . . . . . . . 13 ⊢ ([(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))
63	62	sbcbii 2972	. . . . . . . . . . . 12 ⊢ ([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ [(1st ‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))
64		sbcan 2955	. . . . . . . . . . . 12 ⊢ ([(1st ‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) ↔ ([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))
65		sbcg 2982	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑎) ∈ V → ([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷))
66	38, 65	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷)
67		sbcan 2955	. . . . . . . . . . . . . 14 ⊢ ([(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽) ↔ ([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽))
68		sbceq1g 3027	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑎) ∈ V → ([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1st ‘𝑎) / 𝑥⦌𝑥 = 𝐼))
69	38, 68	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1st ‘𝑎) / 𝑥⦌𝑥 = 𝐼)
70		csbvarg 3035	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑎) ∈ V → ⦋(1st ‘𝑎) / 𝑥⦌𝑥 = (1st ‘𝑎))
71	38, 70	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ⦋(1st ‘𝑎) / 𝑥⦌𝑥 = (1st ‘𝑎)
72	71	eqeq1i 2148	. . . . . . . . . . . . . . . 16 ⊢ (⦋(1st ‘𝑎) / 𝑥⦌𝑥 = 𝐼 ↔ (1st ‘𝑎) = 𝐼)
73	69, 72	bitri 183	. . . . . . . . . . . . . . 15 ⊢ ([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ (1st ‘𝑎) = 𝐼)
74		sbcg 2982	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑎) ∈ V → ([(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽 ↔ (2nd ‘𝑎) = 𝐽))
75	38, 74	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽 ↔ (2nd ‘𝑎) = 𝐽)
76	73, 75	anbi12i 456	. . . . . . . . . . . . . 14 ⊢ (([(1st ‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1st ‘𝑎) / 𝑥](2nd ‘𝑎) = 𝐽) ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))
77	67, 76	bitri 183	. . . . . . . . . . . . 13 ⊢ ([(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽) ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))
78	66, 77	anbi12i 456	. . . . . . . . . . . 12 ⊢ (([(1st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))
79	63, 64, 78	3bitri 205	. . . . . . . . . . 11 ⊢ ([(1st ‘𝑎) / 𝑥][(2nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))
80	18, 46, 79	3bitr3g 221	. . . . . . . . . 10 ⊢ (𝜑 → ((((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))))
81	80	anbi2d 460	. . . . . . . . 9 ⊢ (𝜑 → ((𝑎 ∈ (V × V) ∧ (((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))))
82	15, 81	syl5bb 191	. . . . . . . 8 ⊢ (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))))
83		xpss 4655	. . . . . . . . . . . 12 ⊢ (𝑋 × 𝑌) ⊆ (V × V)
84		simprr 522	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 = ⟨𝐼, 𝐽⟩)
85	8	adantrr 471	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
86	84, 85	eqeltrd 2217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 ∈ (𝑋 × 𝑌))
87	83, 86	sseldi 3100	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 ∈ (V × V))
88	87	ex 114	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩) → 𝑎 ∈ (V × V)))
89	88	pm4.71rd 392	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩))))
90		eqop 6083	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (V × V) → (𝑎 = ⟨𝐼, 𝐽⟩ ↔ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽)))
91	90	anbi2d 460	. . . . . . . . . 10 ⊢ (𝑎 ∈ (V × V) → ((𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩) ↔ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))))
92	91	pm5.32i 450	. . . . . . . . 9 ⊢ ((𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))))
93	89, 92	syl6rbb 196	. . . . . . . 8 ⊢ (𝜑 → ((𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1st ‘𝑎) = 𝐼 ∧ (2nd ‘𝑎) = 𝐽))) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)))
94	82, 93	bitrd 187	. . . . . . 7 ⊢ (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) ∈ 𝐴 ∧ (2nd ‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)))
95	14, 94	syl5bb 191	. . . . . 6 ⊢ (𝜑 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)))
96	95	opabbidv 4002	. . . . 5 ⊢ (𝜑 → {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶)} = {⟨𝑧, 𝑎⟩ ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)})
97		df-mpo 5787	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
98	3, 97	eqtri 2161	. . . . . . 7 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
99	98	cnveqi 4722	. . . . . 6 ⊢ ◡𝐹 = ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
100		nfv 1509	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎 ∈ (𝐴 × 𝐵)
101		nfcsb1v 3040	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶
102	101	nfeq2 2294	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶
103	100, 102	nfan 1545	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶)
104		nfv 1509	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑎 ∈ (𝐴 × 𝐵)
105		nfcv 2282	. . . . . . . . . 10 ⊢ Ⅎ𝑦(1st ‘𝑎)
106		nfcsb1v 3040	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋(2nd ‘𝑎) / 𝑦⦌𝐶
107	105, 106	nfcsb 3042	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶
108	107	nfeq2 2294	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶
109	104, 108	nfan 1545	. . . . . . 7 ⊢ Ⅎ𝑦(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶)
110		eleq1 2203	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
111		opelxp 4577	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
112	110, 111	syl6bb 195	. . . . . . . 8 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
113		csbopeq1a 6094	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶 = 𝐶)
114	113	eqeq2d 2152	. . . . . . . 8 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = 𝐶))
115	112, 114	anbi12d 465	. . . . . . 7 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))
116		xpss 4655	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
117	116	sseli 3098	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐴 × 𝐵) → 𝑎 ∈ (V × V))
118	117	adantr 274	. . . . . . 7 ⊢ ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶) → 𝑎 ∈ (V × V))
119	103, 109, 115, 118	cnvoprab 6139	. . . . . 6 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶)}
120	99, 119	eqtri 2161	. . . . 5 ⊢ ◡𝐹 = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1st ‘𝑎) / 𝑥⦌⦋(2nd ‘𝑎) / 𝑦⦌𝐶)}
121		df-mpt 3999	. . . . 5 ⊢ (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) = {⟨𝑧, 𝑎⟩ ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)}
122	96, 120, 121	3eqtr4g 2198	. . . 4 ⊢ (𝜑 → ◡𝐹 = (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩))
123	122	fneq1d 5221	. . 3 ⊢ (𝜑 → (◡𝐹 Fn 𝐷 ↔ (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷))
124	12, 123	mpbird 166	. 2 ⊢ (𝜑 → ◡𝐹 Fn 𝐷)
125		dff1o4 5383	. 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ◡𝐹 Fn 𝐷))
126	5, 124, 125	sylanbrc 414	1 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  Vcvv 2689  [wsbc 2913  ⦋csb 3007  ⟨cop 3535  {copab 3996   ↦ cmpt 3997   × cxp 4545  ^◡ccnv 4546   Fn wfn 5126  –1-1-onto→wf1o 5130  ‘cfv 5131  {coprab 5783   ∈ cmpo 5784  1^st c1st 6044  2^nd c2nd 6045

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047

This theorem is referenced by:  oddpwdc  11888