Theorem brfvimex 44022

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 5125	. 2 ⊢ (𝜑 → 𝐴(𝐹‘𝐶)𝐵)
4		brne0 5160	. 2 ⊢ (𝐴(𝐹‘𝐶)𝐵 → (𝐹‘𝐶) ≠ ∅)
5		fvprc 6853	. . 3 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐶) = ∅)
6	5	necon1ai 2953	. 2 ⊢ ((𝐹‘𝐶) ≠ ∅ → 𝐶 ∈ V)
7	3, 4, 6	3syl 18	1 ⊢ (𝜑 → 𝐶 ∈ V)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450  ∅c0 4299   class class class wbr 5110  ‘cfv 6514

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-nul 5264  ax-pr 5390

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522

This theorem is referenced by:  ntrclsbex  44030