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Theorem fveqvfvv 47052
Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 6919), 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 6919 . . . 4 (𝐹𝐴) ∈ V
2 eleq1a 2836 . . . 4 ((𝐹𝐴) ∈ V → (V = (𝐹𝐴) → V ∈ V))
31, 2ax-mp 5 . . 3 (V = (𝐹𝐴) → V ∈ V)
4 vprc 5315 . . . 4 ¬ V ∈ V
54pm2.21i 119 . . 3 (V ∈ V → (𝐹𝐴) = 𝐵)
63, 5syl 17 . 2 (V = (𝐹𝐴) → (𝐹𝐴) = 𝐵)
76eqcoms 2745 1 ((𝐹𝐴) = V → (𝐹𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514  df-fv 6569
This theorem is referenced by:  afvpcfv0  47158
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