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Theorem fveqvfvv 28092
Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 5555), 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 5555 . . . 4  |-  ( F `
 A )  e. 
_V
2 eleq1a 2365 . . . 4  |-  ( ( F `  A )  e.  _V  ->  ( _V  =  ( F `  A )  ->  _V  e.  _V ) )
31, 2ax-mp 8 . . 3  |-  ( _V  =  ( F `  A )  ->  _V  e.  _V )
4 vprc 4168 . . . 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 2299 1  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   ` cfv 5271
This theorem is referenced by:  afvpcfv0  28114
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-fv 5279
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