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Theorem fmpt3d 5624
Description: Domain and codomain 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 5623 . 2 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
3 fmpt3d.1 . . 3 (𝜑𝐹 = (𝑥𝐴𝐵))
43feq1d 5307 . 2 (𝜑 → (𝐹:𝐴𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶))
52, 4mpbird 166 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  cmpt 4026  wf 5167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-fv 5179
This theorem is referenced by:  bj-charfun  13424  bj-charfundc  13425
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