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Theorem opbrop 4362
Description: Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
Hypotheses
Ref Expression
opbrop.1 (((z = A w = B) (v = 𝐶 u = 𝐷)) → (φψ))
opbrop.2 𝑅 = {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) φ))}
Assertion
Ref Expression
opbrop (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (⟨A, B𝑅𝐶, 𝐷⟩ ↔ ψ))
Distinct variable groups:   x,y,z,w,v,u,A   x,B,y,z,w,v,u   x,𝐶,y,z,w,v,u   x,𝐷,y,z,w,v,u   x,𝑆,y,z,w,v,u   φ,x,y   ψ,z,w,v,u
Allowed substitution hints:   φ(z,w,v,u)   ψ(x,y)   𝑅(x,y,z,w,v,u)

Proof of Theorem opbrop
StepHypRef Expression
1 opbrop.1 . . . 4 (((z = A w = B) (v = 𝐶 u = 𝐷)) → (φψ))
21copsex4g 3975 . . 3 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) φ) ↔ ψ))
32anbi2d 437 . 2 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (((⟨A, B (𝑆 × 𝑆) 𝐶, 𝐷 (𝑆 × 𝑆)) zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) φ)) ↔ ((⟨A, B (𝑆 × 𝑆) 𝐶, 𝐷 (𝑆 × 𝑆)) ψ)))
4 elex 2560 . . . 4 (A 𝑆A V)
5 elex 2560 . . . 4 (B 𝑆B V)
6 opexgOLD 3956 . . . 4 ((A V B V) → ⟨A, B V)
74, 5, 6syl2an 273 . . 3 ((A 𝑆 B 𝑆) → ⟨A, B V)
8 elex 2560 . . . 4 (𝐶 𝑆𝐶 V)
9 elex 2560 . . . 4 (𝐷 𝑆𝐷 V)
10 opexgOLD 3956 . . . 4 ((𝐶 V 𝐷 V) → ⟨𝐶, 𝐷 V)
118, 9, 10syl2an 273 . . 3 ((𝐶 𝑆 𝐷 𝑆) → ⟨𝐶, 𝐷 V)
12 eleq1 2097 . . . . . 6 (x = ⟨A, B⟩ → (x (𝑆 × 𝑆) ↔ ⟨A, B (𝑆 × 𝑆)))
1312anbi1d 438 . . . . 5 (x = ⟨A, B⟩ → ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) ↔ (⟨A, B (𝑆 × 𝑆) y (𝑆 × 𝑆))))
14 eqeq1 2043 . . . . . . . 8 (x = ⟨A, B⟩ → (x = ⟨z, w⟩ ↔ ⟨A, B⟩ = ⟨z, w⟩))
1514anbi1d 438 . . . . . . 7 (x = ⟨A, B⟩ → ((x = ⟨z, w y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩)))
1615anbi1d 438 . . . . . 6 (x = ⟨A, B⟩ → (((x = ⟨z, w y = ⟨v, u⟩) φ) ↔ ((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) φ)))
17164exbidv 1747 . . . . 5 (x = ⟨A, B⟩ → (zwvu((x = ⟨z, w y = ⟨v, u⟩) φ) ↔ zwvu((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) φ)))
1813, 17anbi12d 442 . . . 4 (x = ⟨A, B⟩ → (((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) φ)) ↔ ((⟨A, B (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) φ))))
19 eleq1 2097 . . . . . 6 (y = ⟨𝐶, 𝐷⟩ → (y (𝑆 × 𝑆) ↔ ⟨𝐶, 𝐷 (𝑆 × 𝑆)))
2019anbi2d 437 . . . . 5 (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B (𝑆 × 𝑆) y (𝑆 × 𝑆)) ↔ (⟨A, B (𝑆 × 𝑆) 𝐶, 𝐷 (𝑆 × 𝑆))))
21 eqeq1 2043 . . . . . . . 8 (y = ⟨𝐶, 𝐷⟩ → (y = ⟨v, u⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩))
2221anbi2d 437 . . . . . . 7 (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩)))
2322anbi1d 438 . . . . . 6 (y = ⟨𝐶, 𝐷⟩ → (((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) φ) ↔ ((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) φ)))
24234exbidv 1747 . . . . 5 (y = ⟨𝐶, 𝐷⟩ → (zwvu((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) φ) ↔ zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) φ)))
2520, 24anbi12d 442 . . . 4 (y = ⟨𝐶, 𝐷⟩ → (((⟨A, B (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) φ)) ↔ ((⟨A, B (𝑆 × 𝑆) 𝐶, 𝐷 (𝑆 × 𝑆)) zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) φ))))
26 opbrop.2 . . . 4 𝑅 = {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) φ))}
2718, 25, 26brabg 3997 . . 3 ((⟨A, B V 𝐶, 𝐷 V) → (⟨A, B𝑅𝐶, 𝐷⟩ ↔ ((⟨A, B (𝑆 × 𝑆) 𝐶, 𝐷 (𝑆 × 𝑆)) zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) φ))))
287, 11, 27syl2an 273 . 2 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (⟨A, B𝑅𝐶, 𝐷⟩ ↔ ((⟨A, B (𝑆 × 𝑆) 𝐶, 𝐷 (𝑆 × 𝑆)) zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) φ))))
29 opelxpi 4319 . . . 4 ((A 𝑆 B 𝑆) → ⟨A, B (𝑆 × 𝑆))
30 opelxpi 4319 . . . 4 ((𝐶 𝑆 𝐷 𝑆) → ⟨𝐶, 𝐷 (𝑆 × 𝑆))
3129, 30anim12i 321 . . 3 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (⟨A, B (𝑆 × 𝑆) 𝐶, 𝐷 (𝑆 × 𝑆)))
3231biantrurd 289 . 2 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (ψ ↔ ((⟨A, B (𝑆 × 𝑆) 𝐶, 𝐷 (𝑆 × 𝑆)) ψ)))
333, 28, 323bitr4d 209 1 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (⟨A, B𝑅𝐶, 𝐷⟩ ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  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:  ecopoveq  6137  oviec  6148
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