Theorem fmptcof 7119

Description: Version of fmptco 7118 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 3898	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅
3	2	nfel1 2915	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵
4		csbeq1a 3888	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑅 = ⦋𝑧 / 𝑥⦌𝑅)
5	4	eleq1d 2819	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑅 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵))
6	3, 5	rspc 3589	. . . . 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 5223	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅)
11	8, 10	eqtrdi 2786	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅))
12		fmptcof.3	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
13		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑤𝑆
14		nfcsb1v 3898	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑆
15		csbeq1a 3888	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝑆 = ⦋𝑤 / 𝑦⦌𝑆)
16	13, 14, 15	cbvmpt 5223	. . . . 5 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆)
17	12, 16	eqtrdi 2786	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆))
18		csbeq1 3877	. . . 4 ⊢ (𝑤 = ⦋𝑧 / 𝑥⦌𝑅 → ⦋𝑤 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
19	7, 11, 17, 18	fmptco 7118	. . 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 3900	. . . 4 ⊢ Ⅎ𝑥⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆
23	4	csbeq1d 3878	. . . 4 ⊢ (𝑥 = 𝑧 → ⦋𝑅 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
24	20, 22, 23	cbvmpt 5223	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
25	19, 24	eqtr4di 2788	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))
26		eqid 2735	. . . 4 ⊢ 𝐴 = 𝐴
27		nfcvd 2899	. . . . . 6 ⊢ (𝑅 ∈ 𝐵 → Ⅎ𝑦𝑇)
28		fmptcof.4	. . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
29	27, 28	csbiegf 3907	. . . . 5 ⊢ (𝑅 ∈ 𝐵 → ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
30	29	ralimi 3073	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
31		mpteq12 5208	. . . 4 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇))
32	26, 30, 31	sylancr 587	. . 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 2770	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ⦋csb 3874   ↦ cmpt 5201   ∘ ccom 5658

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-fv 6538

This theorem is referenced by:  fmptcos  7120  yonedalem3b  18289  gsumcom2  19954  evl1sca  22270  cnmptk1  23617  cnmpt1k  23618  cnmptkk  23619  cncfcompt2  24850  cncfmpt1f  24856  copco  24967  pcoass  24973  sincn  26404  coscn  26405  lgseisenlem3  27338  fcomptf  32582  eulerpartgbij  34350