Theorem fmptcof 7083

Description: Version of fmptco 7082 where 𝜑 needn't be distinct from 𝑥. (Contributed by NM, 27-Dec-2014.)

Hypotheses

Ref	Expression
fmptcof.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵)
fmptcof.2	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
fmptcof.3	⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
fmptcof.4	⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)

Assertion

Ref	Expression
fmptcof	⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

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

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

Proof of Theorem fmptcof

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptcof.1	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵)
2		nfcsb1v 3861	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅
3	2	nfel1 2915	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵
4		csbeq1a 3851	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑅 = ⦋𝑧 / 𝑥⦌𝑅)
5	4	eleq1d 2821	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑅 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵))
6	3, 5	rspc 3552	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵))
7	1, 6	mpan9 506	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵)
8		fmptcof.2	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
9		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑧𝑅
10	9, 2, 4	cbvmpt 5187	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅)
11	8, 10	eqtrdi 2787	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅))
12		fmptcof.3	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
13		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑤𝑆
14		nfcsb1v 3861	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑆
15		csbeq1a 3851	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝑆 = ⦋𝑤 / 𝑦⦌𝑆)
16	13, 14, 15	cbvmpt 5187	. . . . 5 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆)
17	12, 16	eqtrdi 2787	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆))
18		csbeq1 3840	. . . 4 ⊢ (𝑤 = ⦋𝑧 / 𝑥⦌𝑅 → ⦋𝑤 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
19	7, 11, 17, 18	fmptco 7082	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆))
20		nfcv 2898	. . . 4 ⊢ Ⅎ𝑧⦋𝑅 / 𝑦⦌𝑆
21		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝑆
22	2, 21	nfcsbw 3863	. . . 4 ⊢ Ⅎ𝑥⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆
23	4	csbeq1d 3841	. . . 4 ⊢ (𝑥 = 𝑧 → ⦋𝑅 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
24	20, 22, 23	cbvmpt 5187	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
25	19, 24	eqtr4di 2789	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))
26		eqid 2736	. . . 4 ⊢ 𝐴 = 𝐴
27		nfcvd 2899	. . . . . 6 ⊢ (𝑅 ∈ 𝐵 → Ⅎ𝑦𝑇)
28		fmptcof.4	. . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
29	27, 28	csbiegf 3870	. . . . 5 ⊢ (𝑅 ∈ 𝐵 → ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
30	29	ralimi 3074	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
31		mpteq12 5173	. . . 4 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇))
32	26, 30, 31	sylancr 588	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇))
33	1, 32	syl 17	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇))
34	25, 33	eqtrd 2771	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ⦋csb 3837   ↦ cmpt 5166   ∘ ccom 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506

This theorem is referenced by:  fmptcos  7084  yonedalem3b  18245  gsumcom2  19950  evl1sca  22299  cnmptk1  23646  cnmpt1k  23647  cnmptkk  23648  cncfcompt2  24875  cncfmpt1f  24881  copco  24985  pcoass  24991  sincn  26409  coscn  26410  lgseisenlem3  27340  fcomptf  32731  eulerpartgbij  34516