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Theorem fvmptnf 5617
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 5618 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 5169 . . . 4  |-  dom  F  C_  D
32sseli 3176 . . 3  |-  ( A  e.  dom  F  ->  A  e.  D )
4 eqid 2283 . . . . . . 7  |-  ( x  e.  D  |->  (  _I 
`  B ) )  =  ( x  e.  D  |->  (  _I  `  B ) )
51, 4fvmptex 5610 . . . . . 6  |-  ( F `
 A )  =  ( ( x  e.  D  |->  (  _I  `  B ) ) `  A )
6 fvex 5539 . . . . . . 7  |-  (  _I 
`  C )  e. 
_V
7 fvmptf.1 . . . . . . . 8  |-  F/_ x A
8 nfcv 2419 . . . . . . . . 9  |-  F/_ x  _I
9 fvmptf.2 . . . . . . . . 9  |-  F/_ x C
108, 9nffv 5532 . . . . . . . 8  |-  F/_ x
(  _I  `  C
)
11 fvmptf.3 . . . . . . . . 9  |-  ( x  =  A  ->  B  =  C )
1211fveq2d 5529 . . . . . . . 8  |-  ( x  =  A  ->  (  _I  `  B )  =  (  _I  `  C
) )
137, 10, 12, 4fvmptf 5616 . . . . . . 7  |-  ( ( A  e.  D  /\  (  _I  `  C )  e.  _V )  -> 
( ( x  e.  D  |->  (  _I  `  B ) ) `  A )  =  (  _I  `  C ) )
146, 13mpan2 652 . . . . . 6  |-  ( A  e.  D  ->  (
( x  e.  D  |->  (  _I  `  B
) ) `  A
)  =  (  _I 
`  C ) )
155, 14syl5eq 2327 . . . . 5  |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C
) )
16 fvprc 5519 . . . . 5  |-  ( -.  C  e.  _V  ->  (  _I  `  C )  =  (/) )
1715, 16sylan9eq 2335 . . . 4  |-  ( ( A  e.  D  /\  -.  C  e.  _V )  ->  ( F `  A )  =  (/) )
1817expcom 424 . . 3  |-  ( -.  C  e.  _V  ->  ( A  e.  D  -> 
( F `  A
)  =  (/) ) )
193, 18syl5 28 . 2  |-  ( -.  C  e.  _V  ->  ( A  e.  dom  F  ->  ( F `  A
)  =  (/) ) )
20 ndmfv 5552 . 2  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
2119, 20pm2.61d1 151 1  |-  ( -.  C  e.  _V  ->  ( F `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788   (/)c0 3455    e. cmpt 4077    _I cid 4304   dom cdm 4689   ` cfv 5255
This theorem is referenced by:  fvmptn  5618  rdgsucmptnf  6442  frsucmptn  6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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