Theorem mposn 8128

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 7159	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
2	1	3adant3 1133	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
3	2	mpteq1d 5237	. 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 8090	. . . 4 ⊢ (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶) = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶)
6	4, 5	eqtri 2765	. . 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 8027	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9		fveq2 6906	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝐴, 𝐵⟩))
10	9	eqcomd 2743	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘⟨𝐴, 𝐵⟩) = (2nd ‘𝑝))
11	10	eqeq1d 2739	. . . . . . 7 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐵 ↔ (2nd ‘𝑝) = 𝐵))
12	8, 11	syl5ibcom 245	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑝) = 𝐵))
13	12	3adant3 1133	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑝) = 𝐵))
14	13	imp 406	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (2nd ‘𝑝) = 𝐵)
15		op1stg 8026	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
16		fveq2 6906	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑝) = (1st ‘⟨𝐴, 𝐵⟩))
17	16	eqcomd 2743	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘⟨𝐴, 𝐵⟩) = (1st ‘𝑝))
18	17	eqeq1d 2739	. . . . . . 7 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐴 ↔ (1st ‘𝑝) = 𝐴))
19	15, 18	syl5ibcom 245	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑝) = 𝐴))
20	19	3adant3 1133	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑝) = 𝐴))
21	20	imp 406	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st ‘𝑝) = 𝐴)
22		simp1 1137	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐴 ∈ 𝑉)
23		simpl2 1193	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊)
24		mposn.a	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
25	24	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
26		mposn.b	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → 𝐷 = 𝐸)
27	25, 26	sylan9eq 2797	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐸)
28	23, 27	csbied 3935	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑥 = 𝐴) → ⦋𝐵 / 𝑦⦌𝐶 = 𝐸)
29	22, 28	csbied 3935	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐸)
30	29	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐸)
31		csbeq1 3902	. . . . . . . 8 ⊢ ((1st ‘𝑝) = 𝐴 → ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = ⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶)
32	31	eqeq1d 2739	. . . . . . 7 ⊢ ((1st ‘𝑝) = 𝐴 → (⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸 ↔ ⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸))
33	32	adantl 481	. . . . . 6 ⊢ (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → (⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸 ↔ ⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸))
34		csbeq1 3902	. . . . . . . . 9 ⊢ ((2nd ‘𝑝) = 𝐵 → ⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐶)
35	34	adantr 480	. . . . . . . 8 ⊢ (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → ⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐶)
36	35	csbeq2dv 3906	. . . . . . 7 ⊢ (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → ⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶)
37	36	eqeq1d 2739	. . . . . 6 ⊢ (((2nd ‘𝑝) = 𝐵 ∧ (1st ‘𝑝) = 𝐴) → (⦋𝐴 / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸 ↔ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐸))
38	33, 37	bitrd 279	. . . . 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 247	. . . 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 699	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → ⦋(1st ‘𝑝) / 𝑥⦌⦋(2nd ‘𝑝) / 𝑦⦌𝐶 = 𝐸)
41		opex 5469	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
42	41	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ ∈ V)
43		simp3 1139	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐸 ∈ 𝑈)
44	40, 42, 43	fmptsnd 7189	. 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 2787	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐸 ∈ 𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ⦋csb 3899  {csn 4626  ⟨cop 4632   ↦ cmpt 5225   × cxp 5683  ‘cfv 6561   ∈ cmpo 7433  1^st c1st 8012  2^nd c2nd 8013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015

This theorem is referenced by:  mat1dim0  22479  mat1dimid  22480  mat1dimmul  22482  d1mat2pmat  22745