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 8059	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶)
3	1, 2	eqtri 2783	. 2 ⊢ 𝐹 = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶)
4		xp1st 8016	. . 3 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1st ‘𝑤) ∈ 𝐴)
5		xp2nd 8017	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2nd ‘𝑤) ∈ 𝐵)
6		mpof1o2d.r	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)
7	6	anassrs 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
8	7	ralrimiva 3154	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
9		rspcsbela 4395	. . . . . 6 ⊢ (((2nd ‘𝑤) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
10	5, 8, 9	syl2anr 609	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
11	10	an32s 665	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝐴 × 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
12	11	ralrimiva 3154	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
13		rspcsbela 4395	. . 3 ⊢ (((1st ‘𝑤) ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷) → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ 𝐷)
14	4, 12, 13	syl2an2 699	. 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 5686	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ⟨𝐼, 𝐽⟩ ∈ (𝐴 × 𝐵))
18	5	ad2antrl 741	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (2nd ‘𝑤) ∈ 𝐵)
19		sbceq2g 4376	. . . . 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 3793	. . 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 741	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (1st ‘𝑤) ∈ 𝐴)
23	18	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) → (2nd ‘𝑤) ∈ 𝐵)
24		eqop 8026	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ ((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽)))
25	24	ad2antrl 741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ ((1st ‘𝑤) = 𝐼 ∧ (2nd ‘𝑤) = 𝐽)))
26		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑤) → (𝑥 = 𝐼 ↔ (1st ‘𝑤) = 𝐼))
27		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑤) → (𝑦 = 𝐽 ↔ (2nd ‘𝑤) = 𝐽))
28	26, 27	bi2anan9 650	. . . . . . . . 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 519	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
31	30	anassrs 473	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑤 = ⟨𝐼, 𝐽⟩ ↔ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
32		eleq1 2848	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑥 = (1st ‘𝑤)) → 𝑥 ∈ 𝐴)
35		eleq1 2848	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑦 = (2nd ‘𝑤)) → 𝑦 ∈ 𝐵)
38	34, 37	anim12dan 631	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
39	38	3impb 1132	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
40	39	3adant1r 1196	. . . . . . . . 9 ⊢ (((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
41		simp1r 1217	. . . . . . . . 9 ⊢ (((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → 𝑧 ∈ 𝐷)
42	40, 41	jca 521	. . . . . . . 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 605	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))
45	44	3anassrs 1381	. . . . . 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 3781	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) ∧ 𝑥 = (1st ‘𝑤)) → ([(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑤 = ⟨𝐼, 𝐽⟩))
48	22, 47	sbcied 3781	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐷)) → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝑧 = 𝐶 ↔ 𝑤 = ⟨𝐼, 𝐽⟩))
49		sbceq2g 4376	. . . 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 7663	1 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  [wsbc 3738  ⦋csb 3846  ⟨cop 4589   ↦ cmpt 5185   × cxp 5645  –1-1-onto→wf1o 6526  ‘cfv 6527   ∈ cmpo 7410  1^st c1st 7982  2^nd c2nd 7983

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985

This theorem is used by:  evlselvlem  43538