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Theorem fvmpt2f 6476
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 3696 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3701 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2syl6eq 2815 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 fvmpt2f.0 . . 3 𝑥𝐴
5 nfcv 2907 . . 3 𝑦𝐴
6 nfcv 2907 . . 3 𝑦𝐵
7 nfcsb1v 3709 . . 3 𝑥𝑦 / 𝑥𝐵
8 csbeq1a 3702 . . 3 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
94, 5, 6, 7, 8cbvmptf 4909 . 2 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmptg 6473 1 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wnfc 2894  csb 3693  cmpt 4890  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fv 6078
This theorem is referenced by:  offval2f  7111  fmptcof2  29930  funcnvmpt  29940  esumc  30583
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