Theorem fmpoco 8025

Description: Composition of two functions. Variation of fmptco 7062 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)

Hypotheses

Ref	Expression
fmpoco.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶)
fmpoco.2	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅))
fmpoco.3	⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆))
fmpoco.4	⊢ (𝑧 = 𝑅 → 𝑆 = 𝑇)

Assertion

Ref	Expression
fmpoco	⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑧,𝐶,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑧,𝑅   𝑧,𝑇

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)   𝐵(𝑧)   𝑅(𝑥,𝑦)   𝑆(𝑧)   𝑇(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem fmpoco

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmpoco.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶)
2	1	ralrimivva 3175	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑅 ∈ 𝐶)
3		eqid 2731	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅)
4	3	fmpo 8000	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑅 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅):(𝐴 × 𝐵)⟶𝐶)
5	2, 4	sylib 218	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅):(𝐴 × 𝐵)⟶𝐶)
6		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑢𝑅
7		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑣𝑅
8		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
9		nfcsb1v 3874	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝑅
10	8, 9	nfcsbw 3876	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅
11		nfcsb1v 3874	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅
12		csbeq1a 3864	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → 𝑅 = ⦋𝑢 / 𝑥⦌𝑅)
13		csbeq1a 3864	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ⦋𝑢 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
14	12, 13	sylan9eq 2786	. . . . . . 7 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
15	6, 7, 10, 11, 14	cbvmpo 7440	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
16		vex 3440	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
17		vex 3440	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
18	16, 17	op2ndd 7932	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑤) = 𝑣)
19	18	csbeq1d 3854	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅)
20	16, 17	op1std 7931	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑤) = 𝑢)
21	20	csbeq1d 3854	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑢 / 𝑥⦌𝑅)
22	21	csbeq2dv 3857	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋𝑣 / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
23	19, 22	eqtrd 2766	. . . . . . 7 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
24	23	mpompt 7460	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
25	15, 24	eqtr4i 2757	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅)
26	25	fmpt 7043	. . . 4 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅):(𝐴 × 𝐵)⟶𝐶)
27	5, 26	sylibr 234	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝐴 × 𝐵)⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 ∈ 𝐶)
28		fmpoco.2	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅))
29	28, 25	eqtrdi 2782	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅))
30		fmpoco.3	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆))
31	27, 29, 30	fmptcos 7064	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆))
32	23	csbeq1d 3854	. . . . 5 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆 = ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
33	32	mpompt 7460	. . . 4 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
34		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑢⦋𝑅 / 𝑧⦌𝑆
35		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑣⦋𝑅 / 𝑧⦌𝑆
36		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑥𝑆
37	10, 36	nfcsbw 3876	. . . . 5 ⊢ Ⅎ𝑥⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆
38		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑦𝑆
39	11, 38	nfcsbw 3876	. . . . 5 ⊢ Ⅎ𝑦⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆
40	14	csbeq1d 3854	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ⦋𝑅 / 𝑧⦌𝑆 = ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
41	34, 35, 37, 39, 40	cbvmpo 7440	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⦋𝑅 / 𝑧⦌𝑆) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
42	33, 41	eqtr4i 2757	. . 3 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⦋𝑅 / 𝑧⦌𝑆)
43	1	3impb 1114	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ 𝐶)
44		nfcvd 2895	. . . . . 6 ⊢ (𝑅 ∈ 𝐶 → Ⅎ𝑧𝑇)
45		fmpoco.4	. . . . . 6 ⊢ (𝑧 = 𝑅 → 𝑆 = 𝑇)
46	44, 45	csbiegf 3883	. . . . 5 ⊢ (𝑅 ∈ 𝐶 → ⦋𝑅 / 𝑧⦌𝑆 = 𝑇)
47	43, 46	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⦋𝑅 / 𝑧⦌𝑆 = 𝑇)
48	47	mpoeq3dva 7423	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⦋𝑅 / 𝑧⦌𝑆) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
49	42, 48	eqtrid 2778	. 2 ⊢ (𝜑 → (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
50	31, 49	eqtrd 2766	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ⦋csb 3850  ⟨cop 4582   ↦ cmpt 5172   × cxp 5614   ∘ ccom 5620  ⟶wf 6477  ‘cfv 6481   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922

This theorem is referenced by:  oprabco  8026  evlslem2  22012  txswaphmeolem  23717  xpstopnlem1  23722  stdbdxmet  24428  rrxds  25318  cnre2csqima  33919  cvmlift2lem7  35341