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Theorem brfvopab 7490
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 5154 . . . . . 6 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵))
3 brabv 5573 . . . . . 6 (𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
42, 3biimtrdi 253 . . . . 5 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
54imdistani 568 . . . 4 ((𝑋 ∈ V ∧ 𝐴(𝐹𝑋)𝐵) → (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
6 3anass 1095 . . . 4 ((𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
75, 6sylibr 234 . . 3 ((𝑋 ∈ V ∧ 𝐴(𝐹𝑋)𝐵) → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
87ex 412 . 2 (𝑋 ∈ V → (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
9 fvprc 6898 . . 3 𝑋 ∈ V → (𝐹𝑋) = ∅)
10 breq 5145 . . . 4 ((𝐹𝑋) = ∅ → (𝐴(𝐹𝑋)𝐵𝐴𝐵))
11 br0 5192 . . . . 5 ¬ 𝐴𝐵
1211pm2.21i 119 . . . 4 (𝐴𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
1310, 12biimtrdi 253 . . 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 182 1 (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333   class class class wbr 5143  {copab 5205  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569
This theorem is referenced by:  wlkprop  29629  wlkv  29630  isupwlkg  48053
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