Theorem fmptcos 7064

Description: Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

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

Assertion

Ref	Expression
fmptcos	⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))

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

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

Proof of Theorem fmptcos

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptcof.1	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵)
2		fmptcof.2	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
3		fmptcof.3	. . 3 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
4		nfcv 2894	. . . 4 ⊢ Ⅎ𝑧𝑆
5		nfcsb1v 3869	. . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝑆
6		csbeq1a 3859	. . . 4 ⊢ (𝑦 = 𝑧 → 𝑆 = ⦋𝑧 / 𝑦⦌𝑆)
7	4, 5, 6	cbvmpt 5191	. . 3 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆)
8	3, 7	eqtrdi 2782	. 2 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆))
9		csbeq1 3848	. 2 ⊢ (𝑧 = 𝑅 → ⦋𝑧 / 𝑦⦌𝑆 = ⦋𝑅 / 𝑦⦌𝑆)
10	1, 2, 8, 9	fmptcof 7063	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ⦋csb 3845   ↦ cmpt 5170   ∘ ccom 5618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489

This theorem is referenced by:  fmpoco  8025  gsummptf1o  19875  divcncf  25375  gsummpt2d  33029  gsummptfsf1o  33034  fmpocos  42326