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Theorem fveqvfvv 27657
Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 5682), 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 125). (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 5682 . . . 4  |-  ( F `
 A )  e. 
_V
2 eleq1a 2456 . . . 4  |-  ( ( F `  A )  e.  _V  ->  ( _V  =  ( F `  A )  ->  _V  e.  _V ) )
31, 2ax-mp 8 . . 3  |-  ( _V  =  ( F `  A )  ->  _V  e.  _V )
4 vprc 4282 . . . 4  |-  -.  _V  e.  _V
54pm2.21i 125 . . 3  |-  ( _V  e.  _V  ->  ( F `  A )  =  B )
63, 5syl 16 . 2  |-  ( _V  =  ( F `  A )  ->  ( F `  A )  =  B )
76eqcoms 2390 1  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2899   ` cfv 5394
This theorem is referenced by:  afvpcfv0  27679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-sn 3763  df-pr 3764  df-uni 3958  df-iota 5358  df-fv 5402
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