Theorem brfvopab 7466

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 5114	. . . . . 6 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵))
3		brabv 5538	. . . . . 6 ⊢ (𝐴{⟨𝑦, 𝑧⟩ ∣ 𝜑}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	2, 3	biimtrdi 256	. . . . 5 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
5	4	imdistani 579	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
6		3anass 1111	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑋 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7	5, 6	sylibr 237	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝐴(𝐹‘𝑋)𝐵) → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	ex 418	. 2 ⊢ (𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
9		fvprc 6866	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
10		breq 5105	. . . 4 ⊢ ((𝐹‘𝑋) = ∅ → (𝐴(𝐹‘𝑋)𝐵 ↔ 𝐴∅𝐵))
11		br0 5154	. . . . 5 ⊢ ¬ 𝐴∅𝐵
12	11	pm2.21i 120	. . . 4 ⊢ (𝐴∅𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
13	10, 12	biimtrdi 256	. . 3 ⊢ ((𝐹‘𝑋) = ∅ → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
14	9, 13	syl 18	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V)))
15	8, 14	pm2.61i 184	1 ⊢ (𝐴(𝐹‘𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ∅c0 4279   class class class wbr 5103  {copab 5167  ‘cfv 6528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-iota 6484  df-fv 6536

This theorem is used by:  wlkprop  30111  wlkv  30112  isupwlkg  49151