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Theorem fmptdff 44677
Description: A version of fmptd 7129 using bound-variable hypothesis instead of a distinct variable condition for 𝜑. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Hypotheses
Ref Expression
fmptdff.1 𝑥𝜑
fmptdff.2 𝑥𝐴
fmptdff.3 𝑥𝐶
fmptdff.4 ((𝜑𝑥𝐴) → 𝐵𝐶)
fmptdff.5 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fmptdff (𝜑𝐹:𝐴𝐶)

Proof of Theorem fmptdff
StepHypRef Expression
1 fmptdff.1 . . 3 𝑥𝜑
2 fmptdff.4 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐶)
31, 2ralrimia 3253 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 fmptdff.2 . . 3 𝑥𝐴
5 fmptdff.3 . . 3 𝑥𝐶
6 fmptdff.5 . . 3 𝐹 = (𝑥𝐴𝐵)
74, 5, 6fmptff 44675 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
83, 7sylib 217 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wnf 1777  wcel 2098  wnfc 2879  wral 3058  cmpt 5235  wf 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-fun 6555  df-fn 6556  df-f 6557
This theorem is referenced by: (None)
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