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Theorem fvmpt2f 7017
Description: Value of a function given by the maps-to notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
Hypothesis
Ref Expression
fvmpt2f.0 𝑥𝐴
Assertion
Ref Expression
fvmpt2f ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)

Proof of Theorem fvmpt2f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3911 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3921 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2eqtrdi 2791 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 fvmpt2f.0 . . 3 𝑥𝐴
5 nfcv 2903 . . 3 𝑦𝐴
6 nfcv 2903 . . 3 𝑦𝐵
7 nfcsb1v 3933 . . 3 𝑥𝑦 / 𝑥𝐵
8 csbeq1a 3922 . . 3 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
94, 5, 6, 7, 8cbvmptf 5257 . 2 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmptg 7014 1 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wnfc 2888  csb 3908  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-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-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  offval2f  7712  fmptcof2  32674  funcnvmpt  32684  esumc  34032  fvmpt2df  45218  fvmpt4d  45222  smfpimltxrmptf  46714  smfpimgtxrmptf  46740
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