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Theorem fvmpts 5950
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 3438 . 2  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
2 fvmpts.1 . . 3  |-  F  =  ( x  e.  C  |->  B )
3 nfcv 2629 . . . 4  |-  F/_ y B
4 nfcsb1v 3451 . . . 4  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 3444 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 4537 . . 3  |-  ( x  e.  C  |->  B )  =  ( y  e.  C  |->  [_ y  /  x ]_ B )
72, 6eqtri 2496 . 2  |-  F  =  ( y  e.  C  |-> 
[_ y  /  x ]_ B )
81, 7fvmptg 5946 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 1379    e. wcel 1767   [_csb 3435    |-> cmpt 4505   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594
This theorem is referenced by:  fvmptd  5953  fvmpt2curryd  6997  mptnn0fsupp  12066  mptnn0fsuppr  12068  zsum  13496  pcmpt  14263  issubc  15058  gsummptnn0fz  16802  mptscmfsupp0  17356  gsummoncoe1  18114  fvmptnn04if  19114  prdsdsf  20602  itgparts  22180  dchrisumlema  23398  abfmpeld  27161  abfmpel  27162  prodss  28653  fprodser  28655  fprodefsum  28678  fprodn0  28683  aomclem6  30609  ellimcabssub0  31159  constlimc  31166  cdlemk40  35713
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