MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptnf Unicode version

Theorem fvmptnf 5763
Description: The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 5764 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvmptf.1  |-  F/_ x A
fvmptf.2  |-  F/_ x C
fvmptf.3  |-  ( x  =  A  ->  B  =  C )
fvmptf.4  |-  F  =  ( x  e.  D  |->  B )
Assertion
Ref Expression
fvmptnf  |-  ( -.  C  e.  _V  ->  ( F `  A )  =  (/) )
Distinct variable group:    x, D
Allowed substitution hints:    A( x)    B( x)    C( x)    F( x)

Proof of Theorem fvmptnf
StepHypRef Expression
1 fvmptf.4 . . . . 5  |-  F  =  ( x  e.  D  |->  B )
21dmmptss 5308 . . . 4  |-  dom  F  C_  D
32sseli 3289 . . 3  |-  ( A  e.  dom  F  ->  A  e.  D )
4 eqid 2389 . . . . . . 7  |-  ( x  e.  D  |->  (  _I 
`  B ) )  =  ( x  e.  D  |->  (  _I  `  B ) )
51, 4fvmptex 5756 . . . . . 6  |-  ( F `
 A )  =  ( ( x  e.  D  |->  (  _I  `  B ) ) `  A )
6 fvex 5684 . . . . . . 7  |-  (  _I 
`  C )  e. 
_V
7 fvmptf.1 . . . . . . . 8  |-  F/_ x A
8 nfcv 2525 . . . . . . . . 9  |-  F/_ x  _I
9 fvmptf.2 . . . . . . . . 9  |-  F/_ x C
108, 9nffv 5677 . . . . . . . 8  |-  F/_ x
(  _I  `  C
)
11 fvmptf.3 . . . . . . . . 9  |-  ( x  =  A  ->  B  =  C )
1211fveq2d 5674 . . . . . . . 8  |-  ( x  =  A  ->  (  _I  `  B )  =  (  _I  `  C
) )
137, 10, 12, 4fvmptf 5762 . . . . . . 7  |-  ( ( A  e.  D  /\  (  _I  `  C )  e.  _V )  -> 
( ( x  e.  D  |->  (  _I  `  B ) ) `  A )  =  (  _I  `  C ) )
146, 13mpan2 653 . . . . . 6  |-  ( A  e.  D  ->  (
( x  e.  D  |->  (  _I  `  B
) ) `  A
)  =  (  _I 
`  C ) )
155, 14syl5eq 2433 . . . . 5  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
16 fvprc 5664 . . . . 5  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1715, 16sylan9eq 2441 . . . 4  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (/) )
1817expcom 425 . . 3  |-  ( -.  C  e.  _V  ->  ( A  e.  D  -> 
( F `  A
)  =  (/) ) )
193, 18syl5 30 . 2  |-  ( -.  C  e.  _V  ->  ( A  e.  dom  F  ->  ( F `  A
)  =  (/) ) )
20 ndmfv 5697 . 2  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
2119, 20pm2.61d1 153 1  |-  ( -.  C  e.  _V  ->  ( F `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   F/_wnfc 2512   _Vcvv 2901   (/)c0 3573    e. cmpt 4209    _I cid 4436   dom cdm 4820   ` cfv 5396
This theorem is referenced by:  fvmptn  5764  rdgsucmptnf  6625  frsucmptn  6634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404
  Copyright terms: Public domain W3C validator