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Theorem fvmpt2f 6870
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 3839 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3849 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2eqtrdi 2795 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 fvmpt2f.0 . . 3 𝑥𝐴
5 nfcv 2908 . . 3 𝑦𝐴
6 nfcv 2908 . . 3 𝑦𝐵
7 nfcsb1v 3861 . . 3 𝑥𝑦 / 𝑥𝐵
8 csbeq1a 3850 . . 3 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
94, 5, 6, 7, 8cbvmptf 5187 . 2 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmptg 6867 1 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1541  wcel 2109  wnfc 2888  csb 3836  cmpt 5161  cfv 6430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-iota 6388  df-fun 6432  df-fv 6438
This theorem is referenced by:  offval2f  7539  fmptcof2  30973  funcnvmpt  30983  esumc  31998
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