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Theorem fmptcos 7073
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 → ∀𝑥𝐴 𝑅𝐵)
fmptcof.2 (𝜑𝐹 = (𝑥𝐴𝑅))
fmptcof.3 (𝜑𝐺 = (𝑦𝐵𝑆))
Assertion
Ref Expression
fmptcos (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
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.
StepHypRef Expression
1 fmptcof.1 . 2 (𝜑 → ∀𝑥𝐴 𝑅𝐵)
2 fmptcof.2 . 2 (𝜑𝐹 = (𝑥𝐴𝑅))
3 fmptcof.3 . . 3 (𝜑𝐺 = (𝑦𝐵𝑆))
4 nfcv 2905 . . . 4 𝑧𝑆
5 nfcsb1v 3878 . . . 4 𝑦𝑧 / 𝑦𝑆
6 csbeq1a 3867 . . . 4 (𝑦 = 𝑧𝑆 = 𝑧 / 𝑦𝑆)
74, 5, 6cbvmpt 5214 . . 3 (𝑦𝐵𝑆) = (𝑧𝐵𝑧 / 𝑦𝑆)
83, 7eqtrdi 2792 . 2 (𝜑𝐺 = (𝑧𝐵𝑧 / 𝑦𝑆))
9 csbeq1 3856 . 2 (𝑧 = 𝑅𝑧 / 𝑦𝑆 = 𝑅 / 𝑦𝑆)
101, 2, 8, 9fmptcof 7072 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wral 3062  csb 3853  cmpt 5186  ccom 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501
This theorem is referenced by:  fmpoco  8019  gsummptf1o  19693  divcncf  24757  gsummpt2d  31733
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