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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) → (φψ))
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
StepHypRef 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)
31, 2eqsstri 2969 . . 3 𝑅 ⊆ (𝐶 × 𝐷)
43brel 4335 . 2 (A𝑅B → (A 𝐶 B 𝐷))
5 df-br 3756 . . . 4 (A𝑅B ↔ ⟨A, B 𝑅)
61eleq2i 2101 . . . 4 (⟨A, B 𝑅 ↔ ⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)})
75, 6bitri 173 . . 3 (A𝑅B ↔ ⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)})
8 brab2ga.1 . . . 4 ((x = A y = B) → (φψ))
98opelopab2a 3993 . . 3 ((A 𝐶 B 𝐷) → (⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)} ↔ ψ))
107, 9syl5bb 181 . 2 ((A 𝐶 B 𝐷) → (A𝑅Bψ))
114, 10biadan2 429 1 (A𝑅B ↔ ((A 𝐶 B 𝐷) ψ))
 Colors of variables: wff set class 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-bnd 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  7340  ltxr  8445
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