MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmpt3d Structured version   Visualization version   GIF version

Theorem fmpt3d 6341
Description: Domain and co-domain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)
Hypotheses
Ref Expression
fmpt3d.1 (𝜑𝐹 = (𝑥𝐴𝐵))
fmpt3d.2 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
fmpt3d (𝜑𝐹:𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fmpt3d
StepHypRef Expression
1 fmpt3d.2 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐶)
2 eqid 2621 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
31, 2fmptd 6340 . 2 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
4 fmpt3d.1 . . 3 (𝜑𝐹 = (𝑥𝐴𝐵))
54feq1d 5987 . 2 (𝜑 → (𝐹:𝐴𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶))
63, 5mpbird 247 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cmpt 4673  wf 5843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855
This theorem is referenced by:  fmptco  6351  nmof  22433  ofoprabco  29304  sgnsf  29511  qqhf  29809  indf  29856  esumcocn  29920  ofcf  29943  mbfmcst  30099  dstrvprob  30311  dstfrvclim1  30317  signstf  30420  fsovfd  37785  dssmapnvod  37793  binomcxplemnotnn0  38034  sge0seq  39967  hoicvrrex  40074
  Copyright terms: Public domain W3C validator