MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cofmpt Structured version   Visualization version   GIF version

Theorem cofmpt 6439
Description: Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
Hypotheses
Ref Expression
cofmpt.1 (𝜑𝐹:𝐶𝐷)
cofmpt.2 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
cofmpt (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem cofmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cofmpt.2 . 2 ((𝜑𝑥𝐴) → 𝐵𝐶)
2 eqidd 2652 . 2 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3 cofmpt.1 . . 3 (𝜑𝐹:𝐶𝐷)
43feqmptd 6288 . 2 (𝜑𝐹 = (𝑦𝐶 ↦ (𝐹𝑦)))
5 fveq2 6229 . 2 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
61, 2, 4, 5fmptco 6436 1 (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cmpt 4762  ccom 5147  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  esumcocn  30270  ftc1anclem6  33620
  Copyright terms: Public domain W3C validator