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Theorem fvmptn 5618
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 5600. (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 2419 . 2  |-  F/_ x D
2 nfcv 2419 . 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 5617 1  |-  ( -.  C  e.  _V  ->  ( F `  D )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    e. cmpt 4077   ` cfv 5255
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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