Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brfvimex Structured version   Visualization version   GIF version

Theorem brfvimex 43353
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 5156 . 2 (𝜑𝐴(𝐹𝐶)𝐵)
4 brne0 5191 . 2 (𝐴(𝐹𝐶)𝐵 → (𝐹𝐶) ≠ ∅)
5 fvprc 6877 . . 3 𝐶 ∈ V → (𝐹𝐶) = ∅)
65necon1ai 2962 . 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 2934  Vcvv 3468  c0 4317   class class class wbr 5141  cfv 6537
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 2163  ax-ext 2697  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545
This theorem is referenced by:  ntrclsbex  43361
  Copyright terms: Public domain W3C validator