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Theorem fveqvfvv 27987
Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 5539), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything ( see pm2.21i 123). (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
fveqvfvv  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  B )

Proof of Theorem fveqvfvv
StepHypRef Expression
1 fvex 5539 . . . 4  |-  ( F `
 A )  e. 
_V
2 eleq1a 2352 . . . 4  |-  ( ( F `  A )  e.  _V  ->  ( _V  =  ( F `  A )  ->  _V  e.  _V ) )
31, 2ax-mp 8 . . 3  |-  ( _V  =  ( F `  A )  ->  _V  e.  _V )
4 vprc 4152 . . . 4  |-  -.  _V  e.  _V
54pm2.21i 123 . . 3  |-  ( _V  e.  _V  ->  ( F `  A )  =  B )
63, 5syl 15 . 2  |-  ( _V  =  ( F `  A )  ->  ( F `  A )  =  B )
76eqcoms 2286 1  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   ` cfv 5255
This theorem is referenced by:  afvpcfv0  28009
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-fv 5263
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