Theorem fmptcos 7121

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 2895	. . . 4 ⊢ Ⅎ𝑧𝑆
5		nfcsb1v 3910	. . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝑆
6		csbeq1a 3899	. . . 4 ⊢ (𝑦 = 𝑧 → 𝑆 = ⦋𝑧 / 𝑦⦌𝑆)
7	4, 5, 6	cbvmpt 5249	. . 3 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆)
8	3, 7	eqtrdi 2780	. 2 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐵 ↦ ⦋𝑧 / 𝑦⦌𝑆))
9		csbeq1 3888	. 2 ⊢ (𝑧 = 𝑅 → ⦋𝑧 / 𝑦⦌𝑆 = ⦋𝑅 / 𝑦⦌𝑆)
10	1, 2, 8, 9	fmptcof 7120	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3053  ⦋csb 3885   ↦ cmpt 5221   ∘ ccom 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541

This theorem is referenced by:  fmpoco  8075  gsummptf1o  19868  divcncf  25286  gsummpt2d  32628  fmpocos  41515