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Theorem fvmptn 5815
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 5796. (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 2571 . 2  |-  F/_ x D
2 nfcv 2571 . 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 5814 1  |-  ( -.  C  e.  _V  ->  ( F `  D )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620    e. cmpt 4258   ` cfv 5446
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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