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Theorem fvmpt2bd 45180
Description: Value of a function given by the maps-to notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
fvmpt2bd.1 (𝜑𝐹 = (𝑥𝐴𝐵))
Assertion
Ref Expression
fvmpt2bd ((𝜑𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt2bd
StepHypRef Expression
1 fvmpt2bd.1 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
21fveq1d 6907 . . 3 (𝜑 → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
323ad2ant1 1133 . 2 ((𝜑𝑥𝐴𝐵𝐶) → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4 eqid 2736 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54fvmpt2 7026 . . 3 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
653adant1 1130 . 2 ((𝜑𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
73, 6eqtrd 2776 1 ((𝜑𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2107  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-in 3957  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-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568
This theorem is referenced by: (None)
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