Theorem brabg2 35154

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 5658	. . . 4 ⊢ Rel 𝑅
3	2	brrelex1i 5572	. . 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 5391	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒))
7	6	biimpd 232	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 → 𝜒))
8	7	ex 416	. . . 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 5391	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒))
13	12	exbiri 810	. . . 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 215	1 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   class class class wbr 5030  {copab 5092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526

This theorem is referenced by: (None)