Theorem mposn 8107

Description: An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)

Hypotheses

Ref	Expression
mposn.f	⊢ 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)
mposn.a	⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
mposn.b	⊢ (𝑦 = 𝐵 → 𝐷 = 𝐸)

Assertion

Ref	Expression
mposn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐸,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)   𝐷(𝑥, 𝑦)   𝐹(𝑥, 𝑦)

Proof of Theorem mposn

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsng 7142	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
2	1	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
3	2	mpteq1d 5206	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → (𝑝 ∈ ({𝐴} × {𝐵}) ↦ ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶) = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶))
4		mposn.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)
5		mpompts 8071	. . . 4 ⊢ (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶) = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶)
6	4, 5	eqtri 2789	. . 3 ⊢ 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶)
7	6	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶))
8		op2ndg 8008	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9		fveq2 6888	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝐴, 𝐵⟩))
10	9	eqcomd 2772	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘⟨𝐴, 𝐵⟩) = (2nd ‘𝑝))
11	10	eqeq1d 2768	. . . . . . 7 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐵 ↔ (2nd ‘𝑝) = 𝐵))
12	8, 11	syl5ibcom 248	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑝) = 𝐵))
13	12	3adant3 1150	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑝) = 𝐵))
14	13	imp 412	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (2nd ‘𝑝) = 𝐵)
15		op1stg 8007	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
16		fveq2 6888	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑝) = (1st ‘⟨𝐴, 𝐵⟩))
17	16	eqcomd 2772	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘⟨𝐴, 𝐵⟩) = (1st ‘𝑝))
18	17	eqeq1d 2768	. . . . . . 7 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐴 ↔ (1st ‘𝑝) = 𝐴))
19	15, 18	syl5ibcom 248	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑝) = 𝐴))
20	19	3adant3 1150	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑝) = 𝐴))
21	20	imp 412	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st ‘𝑝) = 𝐴)
22		simp1 1154	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐴 ∈ 𝑉)
23		simpl2 1211	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊)
24		mposn.a	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
25	24	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
26		mposn.b	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → 𝐷 = 𝐸)
27	25, 26	sylan9eq 2821	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐸)
28	23, 27	csbied 3892	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑥 = 𝐴) → ⦋𝐵 / 𝑦⦌𝐶 = 𝐸)
29	22, 28	csbied 3892	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐸)
30	29	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐸)
31		csbeq1 3859	. . . . . . . 8 ⊢ ((1st ‘𝑝) = 𝐴 → ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = ⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶)
32	31	eqeq1d 2768	. . . . . . 7 ⊢ ((1st ‘𝑝) = 𝐴 → (⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸 ↔ ⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸))
33	32	adantl 487	. . . . . 6 ⊢ (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → (⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸 ↔ ⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸))
34		csbeq1 3859	. . . . . . . . 9 ⊢ ((2nd ‘𝑝) = 𝐵 → ⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐶)
35	34	adantr 486	. . . . . . . 8 ⊢ (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → ⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐶)
36	35	csbeq2dv 3863	. . . . . . 7 ⊢ (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → ⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶)
37	36	eqeq1d 2768	. . . . . 6 ⊢ (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → (⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸 ↔ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐸))
38	33, 37	bitrd 282	. . . . 5 ⊢ (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → (⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸 ↔ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐸))
39	30, 38	syl5ibrcom 250	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸))
40	14, 21, 39	mp2and 712	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸)
41		opex 5450	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
42	41	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ ∈ V)
43		simp3 1156	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐸 ∈ 𝑈)
44	40, 42, 43	fmptsnd 7174	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → {⟨⟨𝐴, 𝐵⟩, 𝐸⟩} = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶))
45	3, 7, 44	3eqtr4d 2811	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3458  ⦋csb 3856  {csn 4594  ⟨cop 4600   ↦ cmpt 5197   × cxp 5664  ‘cfv 6543   ∈ cmpo 7425  1^st c1st 7993  2^nd c2nd 7994

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996

This theorem is used by:  mat1dim0  22667  mat1dimid  22668  mat1dimmul  22670  d1mat2pmat  22933  setc1ohomfval  50312  setc1ocofval  50313  diag1f1olem  50352