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Theorem fveqvfvv 46990
Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 6920), 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 119). (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
fveqvfvv ((𝐹𝐴) = V → (𝐹𝐴) = 𝐵)

Proof of Theorem fveqvfvv
StepHypRef Expression
1 fvex 6920 . . . 4 (𝐹𝐴) ∈ V
2 eleq1a 2834 . . . 4 ((𝐹𝐴) ∈ V → (V = (𝐹𝐴) → V ∈ V))
31, 2ax-mp 5 . . 3 (V = (𝐹𝐴) → V ∈ V)
4 vprc 5321 . . . 4 ¬ V ∈ V
54pm2.21i 119 . . 3 (V ∈ V → (𝐹𝐴) = 𝐵)
63, 5syl 17 . 2 (V = (𝐹𝐴) → (𝐹𝐴) = 𝐵)
76eqcoms 2743 1 ((𝐹𝐴) = V → (𝐹𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-fv 6571
This theorem is referenced by:  afvpcfv0  47096
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