Theorem fveqvfvv 47025

Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 6899), 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	⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵)

Proof of Theorem fveqvfvv

Step	Hyp	Ref	Expression
1		fvex 6899	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
2		eleq1a 2828	. . . 4 ⊢ ((𝐹‘𝐴) ∈ V → (V = (𝐹‘𝐴) → V ∈ V))
3	1, 2	ax-mp 5	. . 3 ⊢ (V = (𝐹‘𝐴) → V ∈ V)
4		vprc 5295	. . . 4 ⊢ ¬ V ∈ V
5	4	pm2.21i 119	. . 3 ⊢ (V ∈ V → (𝐹‘𝐴) = 𝐵)
6	3, 5	syl 17	. 2 ⊢ (V = (𝐹‘𝐴) → (𝐹‘𝐴) = 𝐵)
7	6	eqcoms 2742	1 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-sn 4607  df-pr 4609  df-uni 4888  df-iota 6494  df-fv 6549

This theorem is referenced by:  afvpcfv0  47131