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Theorem fvmpt2f 27720
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 3423 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 3428 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2syl6eq 2511 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 fvmpt2f.0 . . 3  |-  F/_ x A
5 nfcv 2616 . . 3  |-  F/_ y A
6 nfcv 2616 . . 3  |-  F/_ y B
7 nfcsb1v 3436 . . 3  |-  F/_ x [_ y  /  x ]_ B
8 csbeq1a 3429 . . 3  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
94, 5, 6, 7, 8cbvmptf 27715 . 2  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
103, 9fvmptg 5929 1  |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   F/_wnfc 2602   [_csb 3420    |-> cmpt 4497   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578
This theorem is referenced by:  fmptcof2  27724  offval2f  27733  funcnvmptOLD  27736  funcnvmpt  27737  esumc  28280
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