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Theorem fmptcos 7151
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 2903 . . . 4 𝑧𝑆
5 nfcsb1v 3933 . . . 4 𝑦𝑧 / 𝑦𝑆
6 csbeq1a 3922 . . . 4 (𝑦 = 𝑧𝑆 = 𝑧 / 𝑦𝑆)
74, 5, 6cbvmpt 5259 . . 3 (𝑦𝐵𝑆) = (𝑧𝐵𝑧 / 𝑦𝑆)
83, 7eqtrdi 2791 . 2 (𝜑𝐺 = (𝑧𝐵𝑧 / 𝑦𝑆))
9 csbeq1 3911 . 2 (𝑧 = 𝑅𝑧 / 𝑦𝑆 = 𝑅 / 𝑦𝑆)
101, 2, 8, 9fmptcof 7150 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  csb 3908  cmpt 5231  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  fmpoco  8119  gsummptf1o  19996  divcncf  25496  gsummpt2d  33035  gsummptfsf1o  33040  fmpocos  42254
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