Theorem fveqvfvv 47582

Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 6869), 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 6869	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
2		eleq1a 2851	. . . 4 ⊢ ((𝐹‘𝐴) ∈ V → (V = (𝐹‘𝐴) → V ∈ V))
3	1, 2	ax-mp 5	. . 3 ⊢ (V = (𝐹‘𝐴) → V ∈ V)
4		vprc 5264	. . . 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 2764	1 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1554   ∈ wcel 2136  Vcvv 3448  ‘cfv 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-nul 5250

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-v 3450  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-sn 4577  df-pr 4579  df-uni 4860  df-iota 6466  df-fv 6518

This theorem is referenced by:  afvpcfv0  47688