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Theorem fmpt3d 6531
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)
21fmpttd 6530 . 2 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
3 fmpt3d.1 . . 3 (𝜑𝐹 = (𝑥𝐴𝐵))
43feq1d 6169 . 2 (𝜑 → (𝐹:𝐴𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶))
52, 4mpbird 247 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cmpt 4864  wf 6026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038
This theorem is referenced by:  fmptco  6542  nmof  22743  ofoprabco  29804  sgnsf  30069  qqhf  30370  indf  30417  esumcocn  30482  ofcf  30505  mbfmcst  30661  dstrvprob  30873  dstfrvclim1  30879  signstf  30983  fsovfd  38830  dssmapnvod  38838  binomcxplemnotnn0  39079  sge0seq  41175  hoicvrrex  41285
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