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Theorem brfvimex 43505
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 5167 . 2 (𝜑𝐴(𝐹𝐶)𝐵)
4 brne0 5202 . 2 (𝐴(𝐹𝐶)𝐵 → (𝐹𝐶) ≠ ∅)
5 fvprc 6894 . . 3 𝐶 ∈ V → (𝐹𝐶) = ∅)
65necon1ai 2965 . 2 ((𝐹𝐶) ≠ ∅ → 𝐶 ∈ V)
73, 4, 63syl 18 1 (𝜑𝐶 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wne 2937  Vcvv 3473  c0 4326   class class class wbr 5152  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561
This theorem is referenced by:  ntrclsbex  43513
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