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Theorem fvmpt2bd 45113
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 6909 . . 3 (𝜑 → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
323ad2ant1 1132 . 2 ((𝜑𝑥𝐴𝐵𝐶) → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4 eqid 2735 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54fvmpt2 7027 . . 3 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
653adant1 1129 . 2 ((𝜑𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
73, 6eqtrd 2775 1 ((𝜑𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  cmpt 5231  cfv 6563
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-uni 4913  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-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by: (None)
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