Theorem brabg2 33510

Description: Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
brabg2.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
brabg2.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
brabg2.3	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
brabg2.4	⊢ (𝜒 → 𝐴 ∈ 𝐶)

Assertion

Ref	Expression
brabg2	⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 ↔ 𝜒))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brabg2

Step	Hyp	Ref	Expression
1		brabg2.3	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	relopabi 5245	. . . 4 ⊢ Rel 𝑅
3	2	brrelexi 5158	. . 3 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
4		brabg2.1	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5		brabg2.2	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
6	4, 5, 1	brabg 4994	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒))
7	6	biimpd 219	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 → 𝜒))
8	7	ex 450	. . . 4 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → 𝜒)))
9	8	com3l 89	. . 3 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → (𝐴 ∈ V → 𝜒)))
10	3, 9	mpdi 45	. 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → 𝜒))
11		brabg2.4	. . 3 ⊢ (𝜒 → 𝐴 ∈ 𝐶)
12	4, 5, 1	brabg 4994	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒))
13	12	exbiri 652	. . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝜒 → 𝐴𝑅𝐵)))
14	13	com3l 89	. . 3 ⊢ (𝐵 ∈ 𝐷 → (𝜒 → (𝐴 ∈ 𝐶 → 𝐴𝑅𝐵)))
15	11, 14	mpdi 45	. 2 ⊢ (𝐵 ∈ 𝐷 → (𝜒 → 𝐴𝑅𝐵))
16	10, 15	impbid 202	1 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   class class class wbr 4653  {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)