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Theorem fvmptmap 7474
Description: Special case of fvmpt 5956 for operator theorems. (Contributed by NM, 27-Nov-2007.)
Hypotheses
Ref Expression
fvmptmap.1  |-  C  e. 
_V
fvmptmap.2  |-  D  e. 
_V
fvmptmap.3  |-  R  e. 
_V
fvmptmap.4  |-  ( x  =  A  ->  B  =  C )
fvmptmap.5  |-  F  =  ( x  e.  ( R  ^m  D ) 
|->  B )
Assertion
Ref Expression
fvmptmap  |-  ( A : D --> R  -> 
( F `  A
)  =  C )
Distinct variable groups:    x, A    x, C    x, D    x, R
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmptmap
StepHypRef Expression
1 fvmptmap.3 . . 3  |-  R  e. 
_V
2 fvmptmap.2 . . 3  |-  D  e. 
_V
31, 2elmap 7466 . 2  |-  ( A  e.  ( R  ^m  D )  <->  A : D
--> R )
4 fvmptmap.4 . . 3  |-  ( x  =  A  ->  B  =  C )
5 fvmptmap.5 . . 3  |-  F  =  ( x  e.  ( R  ^m  D ) 
|->  B )
6 fvmptmap.1 . . 3  |-  C  e. 
_V
74, 5, 6fvmpt 5956 . 2  |-  ( A  e.  ( R  ^m  D )  ->  ( F `  A )  =  C )
83, 7sylbir 213 1  |-  ( A : D --> R  -> 
( F `  A
)  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440
This theorem is referenced by:  itg2val  22260  nmopval  26901  nmfnval  26921  eigvecval  26941  eigvalfval  26942  specval  26943
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