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Theorem fvmptnf 5982
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 5983 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 5338 . . . 4  |-  dom  F  C_  D
32sseli 3414 . . 3  |-  ( A  e.  dom  F  ->  A  e.  D )
4 eqid 2471 . . . . . . 7  |-  ( x  e.  D  |->  (  _I 
`  B ) )  =  ( x  e.  D  |->  (  _I  `  B ) )
51, 4fvmptex 5975 . . . . . 6  |-  ( F `
 A )  =  ( ( x  e.  D  |->  (  _I  `  B ) ) `  A )
6 fvex 5889 . . . . . . 7  |-  (  _I 
`  C )  e. 
_V
7 fvmptf.1 . . . . . . . 8  |-  F/_ x A
8 nfcv 2612 . . . . . . . . 9  |-  F/_ x  _I
9 fvmptf.2 . . . . . . . . 9  |-  F/_ x C
108, 9nffv 5886 . . . . . . . 8  |-  F/_ x
(  _I  `  C
)
11 fvmptf.3 . . . . . . . . 9  |-  ( x  =  A  ->  B  =  C )
1211fveq2d 5883 . . . . . . . 8  |-  ( x  =  A  ->  (  _I  `  B )  =  (  _I  `  C
) )
137, 10, 12, 4fvmptf 5981 . . . . . . 7  |-  ( ( A  e.  D  /\  (  _I  `  C )  e.  _V )  -> 
( ( x  e.  D  |->  (  _I  `  B ) ) `  A )  =  (  _I  `  C ) )
146, 13mpan2 685 . . . . . 6  |-  ( A  e.  D  ->  (
( x  e.  D  |->  (  _I  `  B
) ) `  A
)  =  (  _I 
`  C ) )
155, 14syl5eq 2517 . . . . 5  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
16 fvprc 5873 . . . . 5  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1715, 16sylan9eq 2525 . . . 4  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (/) )
1817expcom 442 . . 3  |-  ( -.  C  e.  _V  ->  ( A  e.  D  -> 
( F `  A
)  =  (/) ) )
193, 18syl5 32 . 2  |-  ( -.  C  e.  _V  ->  ( A  e.  dom  F  ->  ( F `  A
)  =  (/) ) )
20 ndmfv 5903 . 2  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
2119, 20pm2.61d1 164 1  |-  ( -.  C  e.  _V  ->  ( F `  A )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904   F/_wnfc 2599   _Vcvv 3031   (/)c0 3722    |-> cmpt 4454    _I cid 4749   dom cdm 4839   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597
This theorem is referenced by:  fvmptn  5983  rdgsucmptnf  7165  frsucmptn  7174
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