Theorem fmptcof 5875

Description: Version of fmptco 5874 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 3180	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅
3	2	nfel1 2403	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵
4		csbeq1a 3156	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑅 = ⦋𝑧 / 𝑥⦌𝑅)
5	4	eleq1d 2307	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑅 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵))
6	3, 5	rspc 2923	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵))
7	1, 6	mpan9 281	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵)
8		fmptcof.2	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
9		nfcv 2392	. . . . . 6 ⊢ Ⅎ𝑧𝑅
10	9, 2, 4	cbvmpt 4226	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅)
11	8, 10	eqtrdi 2287	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅))
12		fmptcof.3	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
13		nfcv 2392	. . . . . 6 ⊢ Ⅎ𝑤𝑆
14		nfcsb1v 3180	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑆
15		csbeq1a 3156	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝑆 = ⦋𝑤 / 𝑦⦌𝑆)
16	13, 14, 15	cbvmpt 4226	. . . . 5 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆)
17	12, 16	eqtrdi 2287	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆))
18		csbeq1 3150	. . . 4 ⊢ (𝑤 = ⦋𝑧 / 𝑥⦌𝑅 → ⦋𝑤 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
19	7, 11, 17, 18	fmptco 5874	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆))
20		nfcv 2392	. . . 4 ⊢ Ⅎ𝑧⦋𝑅 / 𝑦⦌𝑆
21		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑥𝑆
22	2, 21	nfcsb 3185	. . . 4 ⊢ Ⅎ𝑥⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆
23	4	csbeq1d 3154	. . . 4 ⊢ (𝑥 = 𝑧 → ⦋𝑅 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
24	20, 22, 23	cbvmpt 4226	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
25	19, 24	eqtr4di 2289	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))
26		eqid 2238	. . . 4 ⊢ 𝐴 = 𝐴
27		nfcvd 2393	. . . . . 6 ⊢ (𝑅 ∈ 𝐵 → Ⅎ𝑦𝑇)
28		fmptcof.4	. . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
29	27, 28	csbiegf 3191	. . . . 5 ⊢ (𝑅 ∈ 𝐵 → ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
30	29	ralimi 2613	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
31		mpteq12 4214	. . . 4 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇))
32	26, 30, 31	sylancr 418	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇))
33	1, 32	syl 14	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇))
34	25, 33	eqtrd 2271	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ⦋csb 3147   ↦ cmpt 4192   ∘ ccom 4778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385

This theorem is used by:  fmptcos  5876  cncfmpt1f  15790  sincn  15961  coscn  15962  lgseisenlem3  16357