Theorem fveqvfvv 48315

Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 6921), 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 120). (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 6921	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
2		eleq1a 2859	. . . 4 ⊢ ((𝐹‘𝐴) ∈ V → (V = (𝐹‘𝐴) → V ∈ V))
3	1, 2	ax-mp 5	. . 3 ⊢ (V = (𝐹‘𝐴) → V ∈ V)
4		vprc 5278	. . . 4 ⊢ ¬ V ∈ V
5	4	pm2.21i 120	. . 3 ⊢ (V ∈ V → (𝐹‘𝐴) = 𝐵)
6	3, 5	syl 18	. 2 ⊢ (V = (𝐹‘𝐴) → (𝐹‘𝐴) = 𝐵)
7	6	eqcoms 2771	1 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ‘cfv 6558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591  df-uni 4871  df-iota 6512  df-fv 6566

This theorem is used by:  afvpcfv0  48421