Theorem fmptcof 7002

Description: Version of fmptco 7001 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 3857	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅
3	2	nfel1 2923	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵
4		csbeq1a 3846	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑅 = ⦋𝑧 / 𝑥⦌𝑅)
5	4	eleq1d 2823	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑅 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵))
6	3, 5	rspc 3549	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵))
7	1, 6	mpan9 507	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵)
8		fmptcof.2	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
9		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑧𝑅
10	9, 2, 4	cbvmpt 5185	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅)
11	8, 10	eqtrdi 2794	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅))
12		fmptcof.3	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
13		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑤𝑆
14		nfcsb1v 3857	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑆
15		csbeq1a 3846	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝑆 = ⦋𝑤 / 𝑦⦌𝑆)
16	13, 14, 15	cbvmpt 5185	. . . . 5 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆)
17	12, 16	eqtrdi 2794	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆))
18		csbeq1 3835	. . . 4 ⊢ (𝑤 = ⦋𝑧 / 𝑥⦌𝑅 → ⦋𝑤 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
19	7, 11, 17, 18	fmptco 7001	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆))
20		nfcv 2907	. . . 4 ⊢ Ⅎ𝑧⦋𝑅 / 𝑦⦌𝑆
21		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥𝑆
22	2, 21	nfcsbw 3859	. . . 4 ⊢ Ⅎ𝑥⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆
23	4	csbeq1d 3836	. . . 4 ⊢ (𝑥 = 𝑧 → ⦋𝑅 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
24	20, 22, 23	cbvmpt 5185	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
25	19, 24	eqtr4di 2796	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))
26		eqid 2738	. . . 4 ⊢ 𝐴 = 𝐴
27		nfcvd 2908	. . . . . 6 ⊢ (𝑅 ∈ 𝐵 → Ⅎ𝑦𝑇)
28		fmptcof.4	. . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
29	27, 28	csbiegf 3866	. . . . 5 ⊢ (𝑅 ∈ 𝐵 → ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
30	29	ralimi 3087	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
31		mpteq12 5166	. . . 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 2778	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⦋csb 3832   ↦ cmpt 5157   ∘ ccom 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  fmptcos  7003  yonedalem3b  17997  gsumcom2  19576  evl1sca  21500  cnmptk1  22832  cnmpt1k  22833  cnmptkk  22834  cncfcompt2  24071  cncfmpt1f  24077  copco  24181  pcoass  24187  sincn  25603  coscn  25604  lgseisenlem3  26525  fcomptf  30995  eulerpartgbij  32339