Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvmpt2f Structured version   Unicode version

Theorem fvmpt2f 26119
Description: Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
Hypothesis
Ref Expression
fvmpt2f.0  |-  F/_ x A
Assertion
Ref Expression
fvmpt2f  |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )

Proof of Theorem fvmpt2f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3392 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 3397 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2syl6eq 2508 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 fvmpt2f.0 . . 3  |-  F/_ x A
5 nfcv 2613 . . 3  |-  F/_ y A
6 nfcv 2613 . . 3  |-  F/_ y B
7 nfcsb1v 3405 . . 3  |-  F/_ x [_ y  /  x ]_ B
8 csbeq1a 3398 . . 3  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
94, 5, 6, 7, 8cbvmptf 26115 . 2  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
103, 9fvmptg 5874 1  |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   F/_wnfc 2599   [_csb 3389    |-> cmpt 4451   ` cfv 5519
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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527
This theorem is referenced by:  fmptcof2  26123  offval2f  26127  funcnvmptOLD  26130  funcnvmpt  26131  esumc  26643
  Copyright terms: Public domain W3C validator