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Theorem fvmptmap 7052
Description: Special case of fvmpt 5808 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 7044 . 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 5808 . 2  |-  ( A  e.  ( R  ^m  D )  ->  ( F `  A )  =  C )
83, 7sylbir 206 1  |-  ( A : D --> R  -> 
( F `  A
)  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083    ^m cmap 7020
This theorem is referenced by:  itg2val  19622  nmopval  23361  nmfnval  23381  eigvecval  23401  eigvalfval  23402  specval  23403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022
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