Theorem fmpoco 8100

Description: Composition of two functions. Variation of fmptco 7138 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 3197	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑅 ∈ 𝐶)
3		eqid 2728	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅)
4	3	fmpo 8072	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑅 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅):(𝐴 × 𝐵)⟶𝐶)
5	2, 4	sylib 217	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅):(𝐴 × 𝐵)⟶𝐶)
6		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑢𝑅
7		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑣𝑅
8		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
9		nfcsb1v 3917	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝑅
10	8, 9	nfcsbw 3919	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅
11		nfcsb1v 3917	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅
12		csbeq1a 3906	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → 𝑅 = ⦋𝑢 / 𝑥⦌𝑅)
13		csbeq1a 3906	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ⦋𝑢 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
14	12, 13	sylan9eq 2788	. . . . . . 7 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
15	6, 7, 10, 11, 14	cbvmpo 7514	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
16		vex 3475	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
17		vex 3475	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
18	16, 17	op2ndd 8004	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑤) = 𝑣)
19	18	csbeq1d 3896	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅)
20	16, 17	op1std 8003	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑤) = 𝑢)
21	20	csbeq1d 3896	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑢 / 𝑥⦌𝑅)
22	21	csbeq2dv 3899	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋𝑣 / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
23	19, 22	eqtrd 2768	. . . . . . 7 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
24	23	mpompt 7534	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
25	15, 24	eqtr4i 2759	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅)
26	25	fmpt 7120	. . . 4 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅):(𝐴 × 𝐵)⟶𝐶)
27	5, 26	sylibr 233	. . 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 2784	. . 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 7140	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆))
32	23	csbeq1d 3896	. . . . 5 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆 = ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
33	32	mpompt 7534	. . . 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 2899	. . . . 5 ⊢ Ⅎ𝑢⦋𝑅 / 𝑧⦌𝑆
35		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑣⦋𝑅 / 𝑧⦌𝑆
36		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝑆
37	10, 36	nfcsbw 3919	. . . . 5 ⊢ Ⅎ𝑥⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆
38		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑦𝑆
39	11, 38	nfcsbw 3919	. . . . 5 ⊢ Ⅎ𝑦⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆
40	14	csbeq1d 3896	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ⦋𝑅 / 𝑧⦌𝑆 = ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
41	34, 35, 37, 39, 40	cbvmpo 7514	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⦋𝑅 / 𝑧⦌𝑆) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
42	33, 41	eqtr4i 2759	. . 3 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⦋𝑅 / 𝑧⦌𝑆)
43	1	3impb 1113	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ 𝐶)
44		nfcvd 2900	. . . . . 6 ⊢ (𝑅 ∈ 𝐶 → Ⅎ𝑧𝑇)
45		fmpoco.4	. . . . . 6 ⊢ (𝑧 = 𝑅 → 𝑆 = 𝑇)
46	44, 45	csbiegf 3926	. . . . 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 7497	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⦋𝑅 / 𝑧⦌𝑆) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
49	42, 48	eqtrid 2780	. 2 ⊢ (𝜑 → (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
50	31, 49	eqtrd 2768	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3058  ⦋csb 3892  ⟨cop 4635   ↦ cmpt 5231   × cxp 5676   ∘ ccom 5682  ⟶wf 6544  ‘cfv 6548   ∈ cmpo 7422  1^st c1st 7991  2^nd c2nd 7992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994

This theorem is referenced by:  oprabco  8101  evlslem2  22025  txswaphmeolem  23721  xpstopnlem1  23726  stdbdxmet  24437  rrxds  25334  cnre2csqima  33512  cvmlift2lem7  34919