Theorem fmptcof 5331

Description: Version of fmptco 5330 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 2882	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅
3	2	nfel1 2188	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵
4		csbeq1a 2860	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝑅 = ⦋𝑧 / 𝑥⦌𝑅)
5	4	eleq1d 2106	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑅 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵))
6	3, 5	rspc 2650	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵))
7	1, 6	mpan9 265	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵)
8		fmptcof.2	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
9		nfcv 2178	. . . . . 6 ⊢ Ⅎ𝑧𝑅
10	9, 2, 4	cbvmpt 3851	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅)
11	8, 10	syl6eq 2088	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅))
12		fmptcof.3	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
13		nfcv 2178	. . . . . 6 ⊢ Ⅎ𝑤𝑆
14		nfcsb1v 2882	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑆
15		csbeq1a 2860	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝑆 = ⦋𝑤 / 𝑦⦌𝑆)
16	13, 14, 15	cbvmpt 3851	. . . . 5 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆)
17	12, 16	syl6eq 2088	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆))
18		csbeq1 2855	. . . 4 ⊢ (𝑤 = ⦋𝑧 / 𝑥⦌𝑅 → ⦋𝑤 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
19	7, 11, 17, 18	fmptco 5330	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆))
20		nfcv 2178	. . . 4 ⊢ Ⅎ𝑧⦋𝑅 / 𝑦⦌𝑆
21		nfcv 2178	. . . . 5 ⊢ Ⅎ𝑥𝑆
22	2, 21	nfcsb 2884	. . . 4 ⊢ Ⅎ𝑥⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆
23	4	csbeq1d 2858	. . . 4 ⊢ (𝑥 = 𝑧 → ⦋𝑅 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
24	20, 22, 23	cbvmpt 3851	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)
25	19, 24	syl6eqr 2090	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))
26		eqid 2040	. . . 4 ⊢ 𝐴 = 𝐴
27		nfcvd 2179	. . . . . 6 ⊢ (𝑅 ∈ 𝐵 → Ⅎ𝑦𝑇)
28		fmptcof.4	. . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
29	27, 28	csbiegf 2890	. . . . 5 ⊢ (𝑅 ∈ 𝐵 → ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
30	29	ralimi 2384	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇)
31		mpteq12 3840	. . . 4 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇))
32	26, 30, 31	sylancr 393	. . 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 2072	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ⦋csb 2852   ↦ cmpt 3818   ∘ ccom 4349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910

This theorem is referenced by:  fmptcos  5332