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Theorem brfvimex 40383
 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 5083 . 2 (𝜑𝐴(𝐹𝐶)𝐵)
4 brne0 5118 . 2 (𝐴(𝐹𝐶)𝐵 → (𝐹𝐶) ≠ ∅)
5 fvprc 6665 . . 3 𝐶 ∈ V → (𝐹𝐶) = ∅)
65necon1ai 3045 . 2 ((𝐹𝐶) ≠ ∅ → 𝐶 ∈ V)
73, 4, 63syl 18 1 (𝜑𝐶 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  Vcvv 3496  ∅c0 4293   class class class wbr 5068  ‘cfv 6357 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212  ax-pow 5268 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365 This theorem is referenced by:  ntrclsbex  40391
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