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Theorem fvmpt2bd 39870
 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 6334 . . 3 (𝜑 → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
323ad2ant1 1127 . 2 ((𝜑𝑥𝐴𝐵𝐶) → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4 eqid 2771 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54fvmpt2 6433 . . 3 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
653adant1 1124 . 2 ((𝜑𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
73, 6eqtrd 2805 1 ((𝜑𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ↦ cmpt 4863  ‘cfv 6031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039 This theorem is referenced by: (None)
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