MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmpts Structured version   Unicode version

Theorem fvmpts 5886
Description: Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
fvmpts.1  |-  F  =  ( x  e.  C  |->  B )
Assertion
Ref Expression
fvmpts  |-  ( ( A  e.  C  /\  [_ A  /  x ]_ B  e.  V )  ->  ( F `  A
)  =  [_ A  /  x ]_ B )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)    F( x)    V( x)

Proof of Theorem fvmpts
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3399 . 2  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
2 fvmpts.1 . . 3  |-  F  =  ( x  e.  C  |->  B )
3 nfcv 2616 . . . 4  |-  F/_ y B
4 nfcsb1v 3412 . . . 4  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 3405 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 4491 . . 3  |-  ( x  e.  C  |->  B )  =  ( y  e.  C  |->  [_ y  /  x ]_ B )
72, 6eqtri 2483 . 2  |-  F  =  ( y  e.  C  |-> 
[_ y  /  x ]_ B )
81, 7fvmptg 5882 1  |-  ( ( A  e.  C  /\  [_ A  /  x ]_ B  e.  V )  ->  ( F `  A
)  =  [_ A  /  x ]_ B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   [_csb 3396    |-> cmpt 4459   ` cfv 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535
This theorem is referenced by:  fvmptd  5889  fvmpt2curryd  6901  zsum  13314  pcmpt  14073  issubc  14868  mptscmfsupp0  17135  prdsdsf  20075  itgparts  21653  dchrisumlema  22871  abfmpeld  26121  abfmpel  26122  prodss  27605  fprodser  27607  fprodefsum  27630  fprodn0  27635  aomclem6  29561  mptnn0fsupp  30951  mptnn0fsuppr  30953  gsummptnn0fz  30959  gsummoncoe1  30997  fvmptnn04if  31336  cdlemk40  34900
  Copyright terms: Public domain W3C validator