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Theorem fvmpt2i 5607
Description: Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
Hypothesis
Ref Expression
fvmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
fvmpt2i  |-  ( x  e.  A  ->  ( F `  x )  =  (  _I  `  B
) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmpt2i
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3084 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 3088 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2syl6eq 2331 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 fvmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 nfcv 2419 . . . 4  |-  F/_ y B
6 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
85, 6, 7cbvmpt 4110 . . 3  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
94, 8eqtri 2303 . 2  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
103, 9fvmpti 5601 1  |-  ( x  e.  A  ->  ( F `  x )  =  (  _I  `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   [_csb 3081    e. cmpt 4077    _I cid 4304   ` cfv 5255
This theorem is referenced by:  fvmpt2  5608  sumfc  12182  fsumf1o  12196  sumss  12197  isumshft  12298  mbfsup  19019  itg2splitlem  19103  dgrle  19625  prodeq3ii  25308  prodeqfv  25318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263
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