Theorem fveqvfvv 46322

Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 6898), 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 6898	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
2		eleq1a 2822	. . . 4 ⊢ ((𝐹‘𝐴) ∈ V → (V = (𝐹‘𝐴) → V ∈ V))
3	1, 2	ax-mp 5	. . 3 ⊢ (V = (𝐹‘𝐴) → V ∈ V)
4		vprc 5308	. . . 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 2734	1 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-uni 4903  df-iota 6489  df-fv 6545

This theorem is referenced by:  afvpcfv0  46426