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Theorem fvmpt2i 5568
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
Dummy variable  y is distinct from all other variables.
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmpt2i
StepHypRef Expression
1 csbeq1 3085 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 3089 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2syl6eq 2332 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 fvmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 nfcv 2420 . . . 4  |-  F/_ y B
6 nfcsb1v 3114 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 csbeq1a 3090 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
85, 6, 7cbvmpt 4111 . . 3  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
94, 8eqtri 2304 . 2  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
103, 9fvmpti 5562 1  |-  ( x  e.  A  ->  ( F `  x )  =  (  _I  `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   [_csb 3082    e. cmpt 4078    _I cid 4303   ` cfv 5221
This theorem is referenced by:  fvmpt2  5569  sumfc  12176  fsumf1o  12190  sumss  12191  isumshft  12292  mbfsup  19013  itg2splitlem  19097  dgrle  19619  prodeq3ii  24707  prodeqfv  24717
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229
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