Theorem brfvimex 40729

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 5045	. 2 ⊢ (𝜑 → 𝐴(𝐹‘𝐶)𝐵)
4		brne0 5080	. 2 ⊢ (𝐴(𝐹‘𝐶)𝐵 → (𝐹‘𝐶) ≠ ∅)
5		fvprc 6638	. . 3 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐶) = ∅)
6	5	necon1ai 3014	. 2 ⊢ ((𝐹‘𝐶) ≠ ∅ → 𝐶 ∈ V)
7	3, 4, 6	3syl 18	1 ⊢ (𝜑 → 𝐶 ∈ V)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441  ∅c0 4243   class class class wbr 5030  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174  ax-pow 5231

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332

This theorem is referenced by:  ntrclsbex  40737