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Theorem fvmptnf 5967
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 5968 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 5331 . . . 4  |-  dom  F  C_  D
32sseli 3428 . . 3  |-  ( A  e.  dom  F  ->  A  e.  D )
4 eqid 2451 . . . . . . 7  |-  ( x  e.  D  |->  (  _I 
`  B ) )  =  ( x  e.  D  |->  (  _I  `  B ) )
51, 4fvmptex 5960 . . . . . 6  |-  ( F `
 A )  =  ( ( x  e.  D  |->  (  _I  `  B ) ) `  A )
6 fvex 5875 . . . . . . 7  |-  (  _I 
`  C )  e. 
_V
7 fvmptf.1 . . . . . . . 8  |-  F/_ x A
8 nfcv 2592 . . . . . . . . 9  |-  F/_ x  _I
9 fvmptf.2 . . . . . . . . 9  |-  F/_ x C
108, 9nffv 5872 . . . . . . . 8  |-  F/_ x
(  _I  `  C
)
11 fvmptf.3 . . . . . . . . 9  |-  ( x  =  A  ->  B  =  C )
1211fveq2d 5869 . . . . . . . 8  |-  ( x  =  A  ->  (  _I  `  B )  =  (  _I  `  C
) )
137, 10, 12, 4fvmptf 5966 . . . . . . 7  |-  ( ( A  e.  D  /\  (  _I  `  C )  e.  _V )  -> 
( ( x  e.  D  |->  (  _I  `  B ) ) `  A )  =  (  _I  `  C ) )
146, 13mpan2 677 . . . . . 6  |-  ( A  e.  D  ->  (
( x  e.  D  |->  (  _I  `  B
) ) `  A
)  =  (  _I 
`  C ) )
155, 14syl5eq 2497 . . . . 5  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
16 fvprc 5859 . . . . 5  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1715, 16sylan9eq 2505 . . . 4  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (/) )
1817expcom 437 . . 3  |-  ( -.  C  e.  _V  ->  ( A  e.  D  -> 
( F `  A
)  =  (/) ) )
193, 18syl5 33 . 2  |-  ( -.  C  e.  _V  ->  ( A  e.  dom  F  ->  ( F `  A
)  =  (/) ) )
20 ndmfv 5889 . 2  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
2119, 20pm2.61d1 163 1  |-  ( -.  C  e.  _V  ->  ( F `  A )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1444    e. wcel 1887   F/_wnfc 2579   _Vcvv 3045   (/)c0 3731    |-> cmpt 4461    _I cid 4744   dom cdm 4834   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590
This theorem is referenced by:  fvmptn  5968  rdgsucmptnf  7147  frsucmptn  7156
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