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Theorem fvmpt2i 5939
Description: Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
Hypothesis
Ref Expression
mptrcl.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 3375 . . 3  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
2 csbid 3380 . . 3  |-  [_ x  /  x ]_ B  =  B
31, 2syl6eq 2459 . 2  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
4 mptrcl.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 nfcv 2564 . . . 4  |-  F/_ y B
6 nfcsb1v 3388 . . . 4  |-  F/_ x [_ y  /  x ]_ B
7 csbeq1a 3381 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
85, 6, 7cbvmpt 4485 . . 3  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
94, 8eqtri 2431 . 2  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
103, 9fvmpti 5930 1  |-  ( x  e.  A  ->  ( F `  x )  =  (  _I  `  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   [_csb 3372    |-> cmpt 4452    _I cid 4732   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fv 5576
This theorem is referenced by:  fvmpt2  5940  sumfc  13678  fsumf1o  13692  sumss  13693  isumshft  13800  prodfc  13902  fprodf1o  13903  mbfsup  22361  itg2splitlem  22445  dgrle  22930
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