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Theorem fvmpt2df 45284
Description: Deduction version of fvmpt2 7026. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
Hypotheses
Ref Expression
fvmpt2df.1 𝑥𝐴
fvmpt2df.2 𝐹 = (𝑥𝐴𝐵)
fvmpt2df.3 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
fvmpt2df ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)

Proof of Theorem fvmpt2df
StepHypRef Expression
1 fvmpt2df.2 . . 3 𝐹 = (𝑥𝐴𝐵)
21fveq1i 6906 . 2 (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥)
3 id 22 . . 3 (𝑥𝐴𝑥𝐴)
4 fvmpt2df.3 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 fvmpt2df.1 . . . 4 𝑥𝐴
65fvmpt2f 7016 . . 3 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
73, 4, 6syl2an2 686 . 2 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
82, 7eqtrid 2788 1 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wnfc 2889  cmpt 5224  cfv 6560
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-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  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-iota 6513  df-fun 6562  df-fv 6568
This theorem is referenced by:  fsupdm  46862
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