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Theorem fmptdff 45283
Description: A version of fmptd 7133 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 3257 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 fmptdff.2 . . 3 𝑥𝐴
5 fmptdff.3 . . 3 𝑥𝐶
6 fmptdff.5 . . 3 𝐹 = (𝑥𝐴𝐵)
74, 5, 6fmptff 45281 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
83, 7sylib 218 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wnf 1782  wcel 2107  wnfc 2889  wral 3060  cmpt 5224  wf 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564
This theorem is referenced by: (None)
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