Theorem fveqvfvv 47744

Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 6894), 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 6894	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
2		eleq1a 2856	. . . 4 ⊢ ((𝐹‘𝐴) ∈ V → (V = (𝐹‘𝐴) → V ∈ V))
3	1, 2	ax-mp 5	. . 3 ⊢ (V = (𝐹‘𝐴) → V ∈ V)
4		vprc 5282	. . . 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 2769	1 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141  Vcvv 3453  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4589  df-pr 4591  df-uni 4872  df-iota 6492  df-fv 6544

This theorem is referenced by:  afvpcfv0  47850