Theorem brfvimex 44050

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 5134	. 2 ⊢ (𝜑 → 𝐴(𝐹‘𝐶)𝐵)
4		brne0 5169	. 2 ⊢ (𝐴(𝐹‘𝐶)𝐵 → (𝐹‘𝐶) ≠ ∅)
5		fvprc 6868	. . 3 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐶) = ∅)
6	5	necon1ai 2959	. 2 ⊢ ((𝐹‘𝐶) ≠ ∅ → 𝐶 ∈ V)
7	3, 4, 6	3syl 18	1 ⊢ (𝜑 → 𝐶 ∈ V)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  Vcvv 3459  ∅c0 4308   class class class wbr 5119  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539

This theorem is referenced by:  ntrclsbex  44058