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Theorem brfvopab 7200
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 5068 . . . . . 6 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵))
3 brabv 5444 . . . . . 6 (𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
42, 3syl6bi 254 . . . . 5 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
54imdistani 569 . . . 4 ((𝑋 ∈ V ∧ 𝐴(𝐹𝑋)𝐵) → (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
6 3anass 1087 . . . 4 ((𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
75, 6sylibr 235 . . 3 ((𝑋 ∈ V ∧ 𝐴(𝐹𝑋)𝐵) → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
87ex 413 . 2 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
9 fvprc 6656 . . 3 𝑋 ∈ V → (𝐹𝑋) = ∅)
10 breq 5059 . . . 4 ((𝐹𝑋) = ∅ → (𝐴(𝐹𝑋)𝐵𝐴𝐵))
11 br0 5106 . . . . 5 ¬ 𝐴𝐵
1211pm2.21i 119 . . . 4 (𝐴𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
1310, 12syl6bi 254 . . 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 183 1 (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  c0 4288   class class class wbr 5057  {copab 5119  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-iota 6307  df-fv 6356
This theorem is referenced by:  wlkprop  27320  wlkv  27321  isupwlkg  43889
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