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Theorem fmptcos 5595
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 2282 . . . 4 𝑧𝑆
5 nfcsb1v 3039 . . . 4 𝑦𝑧 / 𝑦𝑆
6 csbeq1a 3015 . . . 4 (𝑦 = 𝑧𝑆 = 𝑧 / 𝑦𝑆)
74, 5, 6cbvmpt 4030 . . 3 (𝑦𝐵𝑆) = (𝑧𝐵𝑧 / 𝑦𝑆)
83, 7eqtrdi 2189 . 2 (𝜑𝐺 = (𝑧𝐵𝑧 / 𝑦𝑆))
9 csbeq1 3009 . 2 (𝑧 = 𝑅𝑧 / 𝑦𝑆 = 𝑅 / 𝑦𝑆)
101, 2, 8, 9fmptcof 5594 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wcel 1481  wral 2417  csb 3006  cmpt 3996  ccom 4550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138
This theorem is referenced by:  fmpoco  6120
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