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Theorem fmptdff 45217
Description: A version of fmptd 7134 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 3256 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 fmptdff.2 . . 3 𝑥𝐴
5 fmptdff.3 . . 3 𝑥𝐶
6 fmptdff.5 . . 3 𝐹 = (𝑥𝐴𝐵)
74, 5, 6fmptff 45215 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
83, 7sylib 218 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888  wral 3059  cmpt 5231  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by: (None)
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