Theorem brfvimex 44853

Description: If a binary relation holds and the relation is the value of a function, then the argument to that function is a set. (Contributed by RP, 22-May-2021.)

Hypotheses

Ref	Expression
brfvimex.br	⊢ (𝜑 → 𝐴𝑅𝐵)
brfvimex.fv	⊢ (𝜑 → 𝑅 = (𝐹‘𝐶))

Assertion

Ref	Expression
brfvimex	⊢ (𝜑 → 𝐶 ∈ V)

Proof of Theorem brfvimex

Step	Hyp	Ref	Expression
1		brfvimex.fv	. . 3 ⊢ (𝜑 → 𝑅 = (𝐹‘𝐶))
2		brfvimex.br	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
3	1, 2	breqdi 5122	. 2 ⊢ (𝜑 → 𝐴(𝐹‘𝐶)𝐵)
4		brne0 5159	. 2 ⊢ (𝐴(𝐹‘𝐶)𝐵 → (𝐹‘𝐶) ≠ ∅)
5		fvprc 6874	. . 3 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐶) = ∅)
6	5	necon1ai 2984	. 2 ⊢ ((𝐹‘𝐶) ≠ ∅ → 𝐶 ∈ V)
7	3, 4, 6	3syl 19	1 ⊢ (𝜑 → 𝐶 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  Vcvv 3453  ∅c0 4282   class class class wbr 5107  ‘cfv 6537

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545

This theorem is used by:  ntrclsbex  44861