Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fveqvfvv Structured version   Unicode version

Theorem fveqvfvv 32412
Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 5882), 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 131). (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 5882 . . . 4  |-  ( F `
 A )  e. 
_V
2 eleq1a 2540 . . . 4  |-  ( ( F `  A )  e.  _V  ->  ( _V  =  ( F `  A )  ->  _V  e.  _V ) )
31, 2ax-mp 5 . . 3  |-  ( _V  =  ( F `  A )  ->  _V  e.  _V )
4 vprc 4594 . . . 4  |-  -.  _V  e.  _V
54pm2.21i 131 . . 3  |-  ( _V  e.  _V  ->  ( F `  A )  =  B )
63, 5syl 16 . 2  |-  ( _V  =  ( F `  A )  ->  ( F `  A )  =  B )
76eqcoms 2469 1  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-sn 4033  df-pr 4035  df-uni 4252  df-iota 5557  df-fv 5602
This theorem is referenced by:  afvpcfv0  32434
  Copyright terms: Public domain W3C validator