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Theorem brfvopab 6598
 Description: The classes involved in a binary relation of a function value which is an ordered-pair class abstraction are sets. (Contributed by AV, 7-Jan-2021.)
Hypothesis
Ref Expression
brfvopab.1 (𝑋 ∈ V → (𝐹𝑋) = {⟨𝑦, 𝑧⟩ ∣ 𝜑})
Assertion
Ref Expression
brfvopab (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem brfvopab
StepHypRef Expression
1 brfvopab.1 . . . . . . 7 (𝑋 ∈ V → (𝐹𝑋) = {⟨𝑦, 𝑧⟩ ∣ 𝜑})
21breqd 4594 . . . . . 6 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵))
3 brabv 6597 . . . . . 6 (𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
42, 3syl6bi 242 . . . . 5 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
54imdistani 722 . . . 4 ((𝑋 ∈ V ∧ 𝐴(𝐹𝑋)𝐵) → (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
6 3anass 1035 . . . 4 ((𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
75, 6sylibr 223 . . 3 ((𝑋 ∈ V ∧ 𝐴(𝐹𝑋)𝐵) → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
87ex 449 . 2 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
9 fvprc 6097 . . 3 𝑋 ∈ V → (𝐹𝑋) = ∅)
10 breq 4585 . . . 4 ((𝐹𝑋) = ∅ → (𝐴(𝐹𝑋)𝐵𝐴𝐵))
11 br0 4631 . . . . 5 ¬ 𝐴𝐵
1211pm2.21i 115 . . . 4 (𝐴𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
1310, 12syl6bi 242 . . 3 ((𝐹𝑋) = ∅ → (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
149, 13syl 17 . 2 𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
158, 14pm2.61i 175 1 (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   class class class wbr 4583  {copab 4642  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812 This theorem is referenced by:  wlkv  40815  trlis1wlk  40905  PthisTrl  40931  sPthisPth  40932  spthdep  40940  pthdepissPth  40941  isclWlkb  40980  clwlkis1wlk  40981  crctprop  40998  cyclprop  40999  eupthtrli  41370  eupthi  41371  upgreupthi  41376
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