ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fmptcos GIF version

Theorem fmptcos 5542
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 2255 . . . 4 𝑧𝑆
5 nfcsb1v 3001 . . . 4 𝑦𝑧 / 𝑦𝑆
6 csbeq1a 2979 . . . 4 (𝑦 = 𝑧𝑆 = 𝑧 / 𝑦𝑆)
74, 5, 6cbvmpt 3983 . . 3 (𝑦𝐵𝑆) = (𝑧𝐵𝑧 / 𝑦𝑆)
83, 7syl6eq 2163 . 2 (𝜑𝐺 = (𝑧𝐵𝑧 / 𝑦𝑆))
9 csbeq1 2974 . 2 (𝑧 = 𝑅𝑧 / 𝑦𝑆 = 𝑅 / 𝑦𝑆)
101, 2, 8, 9fmptcof 5541 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  wral 2390  csb 2971  cmpt 3949  ccom 4503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089
This theorem is referenced by:  fmpoco  6067
  Copyright terms: Public domain W3C validator