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Theorem fvmpt2f 6241
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 3522 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3527 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2syl6eq 2676 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 fvmpt2f.0 . . 3 𝑥𝐴
5 nfcv 2767 . . 3 𝑦𝐴
6 nfcv 2767 . . 3 𝑦𝐵
7 nfcsb1v 3535 . . 3 𝑥𝑦 / 𝑥𝐵
8 csbeq1a 3528 . . 3 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
94, 5, 6, 7, 8cbvmptf 4713 . 2 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmptg 6238 1 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wnfc 2754  csb 3519  cmpt 4678  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858
This theorem is referenced by:  offval2f  6863  fmptcof2  29290  funcnvmptOLD  29301  funcnvmpt  29302  esumc  29886
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