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Theorem fvmpt2i 5954
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 3438 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 3443 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2syl6eq 2524 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 fvmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 nfcv 2629 . . . 4  |-  F/_ y B
6 nfcsb1v 3451 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 csbeq1a 3444 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
85, 6, 7cbvmpt 4537 . . 3  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
94, 8eqtri 2496 . 2  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
103, 9fvmpti 5947 1  |-  ( x  e.  A  ->  ( F `  x )  =  (  _I  `  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   [_csb 3435    |-> cmpt 4505    _I cid 4790   ` 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-8 1769  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-pow 4625  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-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594
This theorem is referenced by:  fvmpt2  5955  sumfc  13487  fsumf1o  13501  sumss  13502  isumshft  13607  mbfsup  21803  itg2splitlem  21887  dgrle  22372  prodfc  28651  fprodf1o  28652
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