Theorem f1o2d2 41052

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

Hypotheses

Ref	Expression
f1o2d2.f	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
f1o2d2.r	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)
f1o2d2.i	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐼 ∈ 𝐴)
f1o2d2.j	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐽 ∈ 𝐵)
f1o2d2.1	⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐷)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))

Assertion

Ref	Expression
f1o2d2	⊢ (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 f1o2d2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1o2d2.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2		mpompts 8047	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶)
3	1, 2	eqtri 2760	. 2 ⊢ 𝐹 = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶)
4		xp1st 8003	. . 3 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1st ‘𝑤) ∈ 𝐴)
5		xp2nd 8004	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2nd ‘𝑤) ∈ 𝐵)
6		f1o2d2.r	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)
7	6	anassrs 468	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
8	7	ralrimiva 3146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
9		rspcsbela 4434	. . . . . 6 ⊢ (((2nd ‘𝑤) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
10	5, 8, 9	syl2anr 597	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
11	10	an32s 650	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴 × 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
12	11	ralrimiva 3146	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
13		rspcsbela 4434	. . 3 ⊢ (((1st ‘𝑤) ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷) → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
14	4, 12, 13	syl2an2 684	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
15		f1o2d2.i	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐼 ∈ 𝐴)
16		f1o2d2.j	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐽 ∈ 𝐵)
17	15, 16	opelxpd 5713	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ⟨𝐼, 𝐽⟩ ∈ (𝐴 × 𝐵))
18	5	ad2antrl 726	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (2nd ‘𝑤) ∈ 𝐵)
19		sbceq2g 4415	. . . . 5 ⊢ ((2nd ‘𝑤) ∈ 𝐵 → ([(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
20	18, 19	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → ([(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
21	20	sbcbidv 3835	. . 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 726	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (1st ‘𝑤) ∈ 𝐴)
23	18	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) → (2nd ‘𝑤) ∈ 𝐵)
24		eqop 8013	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ ((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽)))
25	24	ad2antrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ ((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽)))
26		eqeq1 2736	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑤) → (𝑥 = 𝐼 ↔ (1st ‘𝑤) = 𝐼))
27		eqeq1 2736	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑤) → (𝑦 = 𝐽 ↔ (2nd ‘𝑤) = 𝐽))
28	26, 27	bi2anan9 637	. . . . . . . . 9 ⊢ ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ ((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽)))
29	28	bicomd 222	. . . . . . . 8 ⊢ ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽) ↔ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
30	25, 29	sylan9bb 510	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
31	30	anassrs 468	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
32		eleq1 2821	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (1st ‘𝑤) → (𝑥 ∈ 𝐴 ↔ (1st ‘𝑤) ∈ 𝐴))
33	4, 32	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝑥 = (1st ‘𝑤) → 𝑥 ∈ 𝐴))
34	33	imp 407	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑥 = (1st ‘𝑤)) → 𝑥 ∈ 𝐴)
35		eleq1 2821	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (2nd ‘𝑤) → (𝑦 ∈ 𝐵 ↔ (2nd ‘𝑤) ∈ 𝐵))
36	5, 35	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝑦 = (2nd ‘𝑤) → 𝑦 ∈ 𝐵))
37	36	imp 407	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑦 = (2nd ‘𝑤)) → 𝑦 ∈ 𝐵)
38	34, 37	anim12dan 619	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
39	38	3impb 1115	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
40	39	3adant1r 1177	. . . . . . . . 9 ⊢ (((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
41		simp1r 1198	. . . . . . . . 9 ⊢ (((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → 𝑧 ∈ 𝐷)
42	40, 41	jca 512	. . . . . . . 8 ⊢ (((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐷))
43		f1o2d2.1	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐷)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))
44	42, 43	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))
45	44	3anassrs 1360	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) ∧ 𝑦 = (2nd ‘𝑤)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))
46	31, 45	bitr2d 279	. . . . 5 ⊢ ((((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑧 = 𝐶 ↔ 𝑤 = ⟨𝐼, 𝐽⟩))
47	23, 46	sbcied 3821	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) → ([(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑤 = ⟨𝐼, 𝐽⟩))
48	22, 47	sbcied 3821	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑤 = ⟨𝐼, 𝐽⟩))
49		sbceq2g 4415	. . . 4 ⊢ ((1st ‘𝑤) ∈ 𝐴 → ([(1st ‘𝑤) / 𝑥]𝑧 = ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
50	22, 49	syl 17	. . 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 308	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ 𝑧 = ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶))
52	3, 14, 17, 51	f1o2d 7656	1 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  [wsbc 3776  ⦋csb 3892  ⟨cop 4633   ↦ cmpt 5230   × cxp 5673  –1-1-onto→wf1o 6539  ‘cfv 6540   ∈ cmpo 7407  1^st c1st 7969  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972

This theorem is referenced by:  evlselvlem  41155