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Theorem fveqvfvv 27945
Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 5734), 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 5734 . . . 4  |-  ( F `
 A )  e. 
_V
2 eleq1a 2504 . . . 4  |-  ( ( F `  A )  e.  _V  ->  ( _V  =  ( F `  A )  ->  _V  e.  _V ) )
31, 2ax-mp 8 . . 3  |-  ( _V  =  ( F `  A )  ->  _V  e.  _V )
4 vprc 4333 . . . 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 2438 1  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   ` cfv 5446
This theorem is referenced by:  afvpcfv0  27967
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
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5410  df-fv 5454
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