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Theorem fvmpt2df 43400
Description: Deduction version of fvmpt2 6956. (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 6840 . 2 (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥)
3 id 22 . . 3 (𝑥𝐴𝑥𝐴)
4 fvmpt2df.3 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
5 fvmpt2df.1 . . . 4 𝑥𝐴
65fvmpt2f 6946 . . 3 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
73, 4, 6syl2an2 684 . 2 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
82, 7eqtrid 2789 1 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wnfc 2885  cmpt 5186  cfv 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501
This theorem is referenced by:  fsupdm  44977
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