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Theorem fmptcos 5705
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 2332 . . . 4 𝑧𝑆
5 nfcsb1v 3105 . . . 4 𝑦𝑧 / 𝑦𝑆
6 csbeq1a 3081 . . . 4 (𝑦 = 𝑧𝑆 = 𝑧 / 𝑦𝑆)
74, 5, 6cbvmpt 4113 . . 3 (𝑦𝐵𝑆) = (𝑧𝐵𝑧 / 𝑦𝑆)
83, 7eqtrdi 2238 . 2 (𝜑𝐺 = (𝑧𝐵𝑧 / 𝑦𝑆))
9 csbeq1 3075 . 2 (𝑧 = 𝑅𝑧 / 𝑦𝑆 = 𝑅 / 𝑦𝑆)
101, 2, 8, 9fmptcof 5704 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  wral 2468  csb 3072  cmpt 4079  ccom 4648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243
This theorem is referenced by:  fmpoco  6241
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