MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmpt2f Structured version   Visualization version   GIF version

Theorem fvmpt2f 6445
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 3677 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3682 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2syl6eq 2810 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 fvmpt2f.0 . . 3 𝑥𝐴
5 nfcv 2902 . . 3 𝑦𝐴
6 nfcv 2902 . . 3 𝑦𝐵
7 nfcsb1v 3690 . . 3 𝑥𝑦 / 𝑥𝐵
8 csbeq1a 3683 . . 3 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
94, 5, 6, 7, 8cbvmptf 4900 . 2 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmptg 6442 1 ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wnfc 2889  csb 3674  cmpt 4881  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057
This theorem is referenced by:  offval2f  7074  fmptcof2  29766  funcnvmptOLD  29776  funcnvmpt  29777  esumc  30422
  Copyright terms: Public domain W3C validator