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Theorem fvmptn 5634
Description: This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class  C it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 5616. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)
Hypotheses
Ref Expression
fvmptn.1  |-  ( x  =  D  ->  B  =  C )
fvmptn.2  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
fvmptn  |-  ( -.  C  e.  _V  ->  ( F `  D )  =  (/) )
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmptn
StepHypRef Expression
1 nfcv 2432 . 2  |-  F/_ x D
2 nfcv 2432 . 2  |-  F/_ x C
3 fvmptn.1 . 2  |-  ( x  =  D  ->  B  =  C )
4 fvmptn.2 . 2  |-  F  =  ( x  e.  A  |->  B )
51, 2, 3, 4fvmptnf 5633 1  |-  ( -.  C  e.  _V  ->  ( F `  D )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468    e. cmpt 4093   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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