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Theorem brfvimex 38843
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 (𝜑𝐴𝑅𝐵)
brfvimex.fv (𝜑𝑅 = (𝐹𝐶))
Assertion
Ref Expression
brfvimex (𝜑𝐶 ∈ V)

Proof of Theorem brfvimex
StepHypRef Expression
1 brfvimex.fv . . 3 (𝜑𝑅 = (𝐹𝐶))
2 brfvimex.br . . 3 (𝜑𝐴𝑅𝐵)
31, 2breqdi 4799 . 2 (𝜑𝐴(𝐹𝐶)𝐵)
4 brne0 4834 . 2 (𝐴(𝐹𝐶)𝐵 → (𝐹𝐶) ≠ ∅)
5 fvprc 6326 . . 3 𝐶 ∈ V → (𝐹𝐶) = ∅)
65necon1ai 2969 . 2 ((𝐹𝐶) ≠ ∅ → 𝐶 ∈ V)
73, 4, 63syl 18 1 (𝜑𝐶 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wne 2942  Vcvv 3349  c0 4061   class class class wbr 4784  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920  ax-pow 4971
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039
This theorem is referenced by:  ntrclsbex  38851
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