Theorem brfvopab 7417

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 → (𝐹‘𝑋) = {⟨𝑦, 𝑧⟩ ∣ 𝜑})

Assertion

Ref	Expression
brfvopab	⊢ (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem brfvopab

Step	Hyp	Ref	Expression
1		brfvopab.1	. . . . . . 7 ⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {⟨𝑦, 𝑧⟩ ∣ 𝜑})
2	1	breqd 5086	. . . . . 6 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵))
3		brabv 5511	. . . . . 6 ⊢ (𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	2, 3	biimtrdi 255	. . . . 5 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
5	4	imdistani 574	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
6		3anass 1101	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7	5, 6	sylibr 236	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	ex 414	. 2 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
9		fvprc 6823	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
10		breq 5077	. . . 4 ⊢ ((𝐹‘𝑋) = ∅ → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴∅𝐵))
11		br0 5124	. . . . 5 ⊢ ¬ 𝐴∅𝐵
12	11	pm2.21i 119	. . . 4 ⊢ (𝐴∅𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
13	10, 12	biimtrdi 255	. . 3 ⊢ ((𝐹‘𝑋) = ∅ → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
14	9, 13	syl 17	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
15	8, 14	pm2.61i 183	1 ⊢ (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  Vcvv 3433  ∅c0 4264   class class class wbr 5075  {copab 5137  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-iota 6445  df-fv 6497

This theorem is referenced by:  wlkprop  29702  wlkv  29703  isupwlkg  48642