Theorem fmptcos 7153

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 2926	. . . 4 ⊢ Ⅎ𝑧𝑆
5		nfcsb1v 3875	. . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝑆
6		csbeq1a 3865	. . . 4 ⊢ (𝑦 = 𝑧 → 𝑆 = ⦋𝑧 / 𝑦⦌𝑆)
7	4, 5, 6	cbvmpt 5210	. . 3 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆)
8	3, 7	eqtrdi 2814	. 2 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆))
9		csbeq1 3854	. 2 ⊢ (𝑧 = 𝑅 → ⦋𝑧 / 𝑦⦌𝑆 = ⦋𝑅 / 𝑦⦌𝑆)
10	1, 2, 8, 9	fmptcof 7152	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⦋csb 3851   ↦ cmpt 5189   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563

This theorem is used by:  fmpoco  8129  gsummptf1o  20204  divcncf  25795  gsummpt2d  33677  gsummptf1od  33683  gsummptfsf1o  33688  fmpocos  43478