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Theorem brfvopab 6654
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 4629 . . . . . 6 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵))
3 brabv 6653 . . . . . 6 (𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
42, 3syl6bi 243 . . . . 5 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
54imdistani 725 . . . 4 ((𝑋 ∈ V ∧ 𝐴(𝐹𝑋)𝐵) → (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
6 3anass 1040 . . . 4 ((𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
75, 6sylibr 224 . . 3 ((𝑋 ∈ V ∧ 𝐴(𝐹𝑋)𝐵) → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
87ex 450 . 2 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
9 fvprc 6144 . . 3 𝑋 ∈ V → (𝐹𝑋) = ∅)
10 breq 4620 . . . 4 ((𝐹𝑋) = ∅ → (𝐴(𝐹𝑋)𝐵𝐴𝐵))
11 br0 4666 . . . . 5 ¬ 𝐴𝐵
1211pm2.21i 116 . . . 4 (𝐴𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
1310, 12syl6bi 243 . . 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 176 1 (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191  c0 3896   class class class wbr 4618  {copab 4677  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-iota 5813  df-fv 5858
This theorem is referenced by:  wlkprop  26371  wlkv  26372  isupwlkg  40994
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