Theorem brfvimex 44607

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 5117	. 2 ⊢ (𝜑 → 𝐴(𝐹‘𝐶)𝐵)
4		brne0 5152	. 2 ⊢ (𝐴(𝐹‘𝐶)𝐵 → (𝐹‘𝐶) ≠ ∅)
5		fvprc 6861	. . 3 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐶) = ∅)
6	5	necon1ai 2986	. 2 ⊢ ((𝐹‘𝐶) ≠ ∅ → 𝐶 ∈ V)
7	3, 4, 6	3syl 18	1 ⊢ (𝜑 → 𝐶 ∈ V)

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  Vcvv 3456  ∅c0 4287   class class class wbr 5102  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531

This theorem is referenced by:  ntrclsbex  44615