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Theorem brab2a 4316
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
Hypotheses
Ref Expression
brab2a.1 ((x = A y = B) → (φψ))
brab2a.2 𝑅 = {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)}
Assertion
Ref Expression
brab2a (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 brab2a
StepHypRef Expression
1 simpl 102 . . . . 5 (((x 𝐶 y 𝐷) φ) → (x 𝐶 y 𝐷))
21ssopab2i 3984 . . . 4 {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)} ⊆ {⟨x, y⟩ ∣ (x 𝐶 y 𝐷)}
3 brab2a.2 . . . 4 𝑅 = {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)}
4 df-xp 4274 . . . 4 (𝐶 × 𝐷) = {⟨x, y⟩ ∣ (x 𝐶 y 𝐷)}
52, 3, 43sstr4i 2957 . . 3 𝑅 ⊆ (𝐶 × 𝐷)
65brel 4315 . 2 (A𝑅B → (A 𝐶 B 𝐷))
7 df-br 3735 . . . 4 (A𝑅B ↔ ⟨A, B 𝑅)
83eleq2i 2082 . . . 4 (⟨A, B 𝑅 ↔ ⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)})
97, 8bitri 173 . . 3 (A𝑅B ↔ ⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)})
10 brab2a.1 . . . 4 ((x = A y = B) → (φψ))
1110opelopab2a 3972 . . 3 ((A 𝐶 B 𝐷) → (⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)} ↔ ψ))
129, 11syl5bb 181 . 2 ((A 𝐶 B 𝐷) → (A𝑅Bψ))
136, 12biadan2 432 1 (A𝑅B ↔ ((A 𝐶 B 𝐷) ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226   wcel 1370  cop 3349   class class class wbr 3734  {copab 3787   × cxp 4266
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274
This theorem is referenced by: (None)
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