Theorem mpof1o2d 8120

Description: Sufficient condition for a binary function expressed in maps-to notation to be bijective. (Contributed by SN, 11-Mar-2025.)

Hypotheses

Ref	Expression
mpof1o2d.f	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
mpof1o2d.r	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)
mpof1o2d.i	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐼 ∈ 𝐴)
mpof1o2d.j	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐽 ∈ 𝐵)
mpof1o2d.1	⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐷)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))

Assertion

Ref	Expression
mpof1o2d	⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)

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

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

Proof of Theorem mpof1o2d

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpof1o2d.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2		mpompts 8061	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶)
3	1, 2	eqtri 2784	. 2 ⊢ 𝐹 = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶)
4		xp1st 8017	. . 3 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1st ‘𝑤) ∈ 𝐴)
5		xp2nd 8018	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2nd ‘𝑤) ∈ 𝐵)
6		mpof1o2d.r	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)
7	6	anassrs 472	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
8	7	ralrimiva 3155	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
9		rspcsbela 4402	. . . . . 6 ⊢ (((2nd ‘𝑤) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
10	5, 8, 9	syl2anr 608	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
11	10	an32s 664	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴 × 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
12	11	ralrimiva 3155	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
13		rspcsbela 4402	. . 3 ⊢ (((1st ‘𝑤) ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷) → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
14	4, 12, 13	syl2an2 698	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
15		mpof1o2d.i	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐼 ∈ 𝐴)
16		mpof1o2d.j	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐽 ∈ 𝐵)
17	15, 16	opelxpd 5700	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ⟨𝐼, 𝐽⟩ ∈ (𝐴 × 𝐵))
18	5	ad2antrl 740	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (2nd ‘𝑤) ∈ 𝐵)
19		sbceq2g 4383	. . . . 5 ⊢ ((2nd ‘𝑤) ∈ 𝐵 → ([(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
20	18, 19	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → ([(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
21	20	sbcbidv 3798	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ [(1st ‘𝑤) / 𝑥]𝑧 = ⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
22	4	ad2antrl 740	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (1st ‘𝑤) ∈ 𝐴)
23	18	adantr 485	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) → (2nd ‘𝑤) ∈ 𝐵)
24		eqop 8027	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ ((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽)))
25	24	ad2antrl 740	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ ((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽)))
26		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑤) → (𝑥 = 𝐼 ↔ (1st ‘𝑤) = 𝐼))
27		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑤) → (𝑦 = 𝐽 ↔ (2nd ‘𝑤) = 𝐽))
28	26, 27	bi2anan9 649	. . . . . . . . 9 ⊢ ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ ((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽)))
29	28	bicomd 226	. . . . . . . 8 ⊢ ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽) ↔ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
30	25, 29	sylan9bb 518	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
31	30	anassrs 472	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
32		eleq1 2849	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (1st ‘𝑤) → (𝑥 ∈ 𝐴 ↔ (1st ‘𝑤) ∈ 𝐴))
33	4, 32	syl5ibrcom 250	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝑥 = (1st ‘𝑤) → 𝑥 ∈ 𝐴))
34	33	imp 411	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑥 = (1st ‘𝑤)) → 𝑥 ∈ 𝐴)
35		eleq1 2849	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (2nd ‘𝑤) → (𝑦 ∈ 𝐵 ↔ (2nd ‘𝑤) ∈ 𝐵))
36	5, 35	syl5ibrcom 250	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝑦 = (2nd ‘𝑤) → 𝑦 ∈ 𝐵))
37	36	imp 411	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑦 = (2nd ‘𝑤)) → 𝑦 ∈ 𝐵)
38	34, 37	anim12dan 630	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
39	38	3impb 1130	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
40	39	3adant1r 1194	. . . . . . . . 9 ⊢ (((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
41		simp1r 1215	. . . . . . . . 9 ⊢ (((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → 𝑧 ∈ 𝐷)
42	40, 41	jca 520	. . . . . . . 8 ⊢ (((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐷))
43		mpof1o2d.1	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐷)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))
44	42, 43	sylan2 604	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))
45	44	3anassrs 1379	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) ∧ 𝑦 = (2nd ‘𝑤)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))
46	31, 45	bitr2d 283	. . . . 5 ⊢ ((((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑧 = 𝐶 ↔ 𝑤 = ⟨𝐼, 𝐽⟩))
47	23, 46	sbcied 3786	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) → ([(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑤 = ⟨𝐼, 𝐽⟩))
48	22, 47	sbcied 3786	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑤 = ⟨𝐼, 𝐽⟩))
49		sbceq2g 4383	. . . 4 ⊢ ((1st ‘𝑤) ∈ 𝐴 → ([(1st ‘𝑤) / 𝑥]𝑧 = ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
50	22, 49	syl 18	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → ([(1st ‘𝑤) / 𝑥]𝑧 = ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
51	21, 48, 50	3bitr3d 312	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ 𝑧 = ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
52	3, 14, 17, 51	f1o2d 7664	1 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∀wral 3077  [wsbc 3743  ⦋csb 3852  ⟨cop 4594   ↦ cmpt 5191   × cxp 5659  –1-1-onto→wf1o 6535  ‘cfv 6536   ∈ cmpo 7412  1^st c1st 7983  2^nd c2nd 7984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986

This theorem is referenced by:  evlselvlem  43290