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Theorem fvmptnf 5814
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 5815 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 5358 . . . 4  |-  dom  F  C_  D
32sseli 3336 . . 3  |-  ( A  e.  dom  F  ->  A  e.  D )
4 eqid 2435 . . . . . . 7  |-  ( x  e.  D  |->  (  _I 
`  B ) )  =  ( x  e.  D  |->  (  _I  `  B ) )
51, 4fvmptex 5807 . . . . . 6  |-  ( F `
 A )  =  ( ( x  e.  D  |->  (  _I  `  B ) ) `  A )
6 fvex 5734 . . . . . . 7  |-  (  _I 
`  C )  e. 
_V
7 fvmptf.1 . . . . . . . 8  |-  F/_ x A
8 nfcv 2571 . . . . . . . . 9  |-  F/_ x  _I
9 fvmptf.2 . . . . . . . . 9  |-  F/_ x C
108, 9nffv 5727 . . . . . . . 8  |-  F/_ x
(  _I  `  C
)
11 fvmptf.3 . . . . . . . . 9  |-  ( x  =  A  ->  B  =  C )
1211fveq2d 5724 . . . . . . . 8  |-  ( x  =  A  ->  (  _I  `  B )  =  (  _I  `  C
) )
137, 10, 12, 4fvmptf 5813 . . . . . . 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 2479 . . . . 5  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
16 fvprc 5714 . . . . 5  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1715, 16sylan9eq 2487 . . . 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 5747 . 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 1652    e. wcel 1725   F/_wnfc 2558   _Vcvv 2948   (/)c0 3620    e. cmpt 4258    _I cid 4485   dom cdm 4870   ` cfv 5446
This theorem is referenced by:  fvmptn  5815  rdgsucmptnf  6679  frsucmptn  6688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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