Theorem brab2ga 4358

Description: The law of concretion for a binary relation. See brab2a 4336 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
brab2ga.1	⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
brab2ga.2	⊢ 𝑅 = {⟨x, y⟩ ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)}

Assertion

Ref	Expression
brab2ga	⊢ (A𝑅B ↔ ((A ∈ 𝐶 ∧ B ∈ 𝐷) ∧ ψ))

Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y   x,𝐷,y   ψ,x,y

Allowed substitution hints:   φ(x,y)   𝑅(x,y)

Proof of Theorem brab2ga

Step	Hyp	Ref	Expression
1		brab2ga.2	. . . 4 ⊢ 𝑅 = {⟨x, y⟩ ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)}
2		opabssxp 4357	. . . 4 ⊢ {⟨x, y⟩ ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)} ⊆ (𝐶 × 𝐷)
3	1, 2	eqsstri 2969	. . 3 ⊢ 𝑅 ⊆ (𝐶 × 𝐷)
4	3	brel 4335	. 2 ⊢ (A𝑅B → (A ∈ 𝐶 ∧ B ∈ 𝐷))
5		df-br 3756	. . . 4 ⊢ (A𝑅B ↔ ⟨A, B⟩ ∈ 𝑅)
6	1	eleq2i 2101	. . . 4 ⊢ (⟨A, B⟩ ∈ 𝑅 ↔ ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)})
7	5, 6	bitri 173	. . 3 ⊢ (A𝑅B ↔ ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)})
8		brab2ga.1	. . . 4 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
9	8	opelopab2a 3993	. . 3 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)} ↔ ψ))
10	7, 9	syl5bb 181	. 2 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (A𝑅B ↔ ψ))
11	4, 10	biadan2 429	1 ⊢ (A𝑅B ↔ ((A ∈ 𝐶 ∧ B ∈ 𝐷) ∧ ψ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ⟨cop 3370   class class class wbr 3755  {copab 3808   × cxp 4286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294

This theorem is referenced by:  reapval  7360  ltxr  8465