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Theorem brecop 6107
Description: Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
Hypotheses
Ref Expression
brecop.1 V
brecop.2 Er (𝐺 × 𝐺)
brecop.4 𝐻 = ((𝐺 × 𝐺) / )
brecop.5 = {⟨x, y⟩ ∣ ((x 𝐻 y 𝐻) zwvu((x = [⟨z, w⟩] y = [⟨v, u⟩] ) φ))}
brecop.6 ((((z 𝐺 w 𝐺) (A 𝐺 B 𝐺)) ((v 𝐺 u 𝐺) (𝐶 𝐺 𝐷 𝐺))) → (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) → (φψ)))
Assertion
Ref Expression
brecop (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → ([⟨A, B⟩] [⟨𝐶, 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, ,y,z,w,v,u   x,𝐻,y   z,𝐺,w,v,u   φ,x,y   ψ,z,w,v,u
Allowed substitution hints:   φ(z,w,v,u)   ψ(x,y)   𝐺(x,y)   𝐻(z,w,v,u)   (x,y,z,w,v,u)

Proof of Theorem brecop
StepHypRef Expression
1 brecop.1 . . . 4 V
2 brecop.4 . . . 4 𝐻 = ((𝐺 × 𝐺) / )
31, 2ecopqsi 6072 . . 3 ((A 𝐺 B 𝐺) → [⟨A, B⟩] 𝐻)
41, 2ecopqsi 6072 . . 3 ((𝐶 𝐺 𝐷 𝐺) → [⟨𝐶, 𝐷⟩] 𝐻)
5 df-br 3739 . . . . 5 ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] ↔ ⟨[⟨A, B⟩] , [⟨𝐶, 𝐷⟩] )
6 brecop.5 . . . . . 6 = {⟨x, y⟩ ∣ ((x 𝐻 y 𝐻) zwvu((x = [⟨z, w⟩] y = [⟨v, u⟩] ) φ))}
76eleq2i 2086 . . . . 5 (⟨[⟨A, B⟩] , [⟨𝐶, 𝐷⟩] ↔ ⟨[⟨A, B⟩] , [⟨𝐶, 𝐷⟩] {⟨x, y⟩ ∣ ((x 𝐻 y 𝐻) zwvu((x = [⟨z, w⟩] y = [⟨v, u⟩] ) φ))})
85, 7bitri 173 . . . 4 ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] ↔ ⟨[⟨A, B⟩] , [⟨𝐶, 𝐷⟩] {⟨x, y⟩ ∣ ((x 𝐻 y 𝐻) zwvu((x = [⟨z, w⟩] y = [⟨v, u⟩] ) φ))})
9 eqeq1 2028 . . . . . . . 8 (x = [⟨A, B⟩] → (x = [⟨z, w⟩] ↔ [⟨A, B⟩] = [⟨z, w⟩] ))
109anbi1d 441 . . . . . . 7 (x = [⟨A, B⟩] → ((x = [⟨z, w⟩] y = [⟨v, u⟩] ) ↔ ([⟨A, B⟩] = [⟨z, w⟩] y = [⟨v, u⟩] )))
1110anbi1d 441 . . . . . 6 (x = [⟨A, B⟩] → (((x = [⟨z, w⟩] y = [⟨v, u⟩] ) φ) ↔ (([⟨A, B⟩] = [⟨z, w⟩] y = [⟨v, u⟩] ) φ)))
12114exbidv 1732 . . . . 5 (x = [⟨A, B⟩] → (zwvu((x = [⟨z, w⟩] y = [⟨v, u⟩] ) φ) ↔ zwvu(([⟨A, B⟩] = [⟨z, w⟩] y = [⟨v, u⟩] ) φ)))
13 eqeq1 2028 . . . . . . . 8 (y = [⟨𝐶, 𝐷⟩] → (y = [⟨v, u⟩] ↔ [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ))
1413anbi2d 440 . . . . . . 7 (y = [⟨𝐶, 𝐷⟩] → (([⟨A, B⟩] = [⟨z, w⟩] y = [⟨v, u⟩] ) ↔ ([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] )))
1514anbi1d 441 . . . . . 6 (y = [⟨𝐶, 𝐷⟩] → ((([⟨A, B⟩] = [⟨z, w⟩] y = [⟨v, u⟩] ) φ) ↔ (([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) φ)))
16154exbidv 1732 . . . . 5 (y = [⟨𝐶, 𝐷⟩] → (zwvu(([⟨A, B⟩] = [⟨z, w⟩] y = [⟨v, u⟩] ) φ) ↔ zwvu(([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) φ)))
1712, 16opelopab2 3981 . . . 4 (([⟨A, B⟩] 𝐻 [⟨𝐶, 𝐷⟩] 𝐻) → (⟨[⟨A, B⟩] , [⟨𝐶, 𝐷⟩] {⟨x, y⟩ ∣ ((x 𝐻 y 𝐻) zwvu((x = [⟨z, w⟩] y = [⟨v, u⟩] ) φ))} ↔ zwvu(([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) φ)))
188, 17syl5bb 181 . . 3 (([⟨A, B⟩] 𝐻 [⟨𝐶, 𝐷⟩] 𝐻) → ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] zwvu(([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) φ)))
193, 4, 18syl2an 273 . 2 (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] zwvu(([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) φ)))
20 opeq12 3525 . . . . . 6 ((z = A w = B) → ⟨z, w⟩ = ⟨A, B⟩)
2120eceq1d 6053 . . . . 5 ((z = A w = B) → [⟨z, w⟩] = [⟨A, B⟩] )
22 opeq12 3525 . . . . . 6 ((v = 𝐶 u = 𝐷) → ⟨v, u⟩ = ⟨𝐶, 𝐷⟩)
2322eceq1d 6053 . . . . 5 ((v = 𝐶 u = 𝐷) → [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] )
2421, 23anim12i 321 . . . 4 (((z = A w = B) (v = 𝐶 u = 𝐷)) → ([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ))
25 opelxpi 4303 . . . . . . . 8 ((A 𝐺 B 𝐺) → ⟨A, B (𝐺 × 𝐺))
26 opelxp 4301 . . . . . . . . 9 (⟨z, w (𝐺 × 𝐺) ↔ (z 𝐺 w 𝐺))
27 brecop.2 . . . . . . . . . . 11 Er (𝐺 × 𝐺)
2827a1i 9 . . . . . . . . . 10 ([⟨z, w⟩] = [⟨A, B⟩] Er (𝐺 × 𝐺))
29 id 19 . . . . . . . . . 10 ([⟨z, w⟩] = [⟨A, B⟩] → [⟨z, w⟩] = [⟨A, B⟩] )
3028, 29ereldm 6060 . . . . . . . . 9 ([⟨z, w⟩] = [⟨A, B⟩] → (⟨z, w (𝐺 × 𝐺) ↔ ⟨A, B (𝐺 × 𝐺)))
3126, 30syl5bbr 183 . . . . . . . 8 ([⟨z, w⟩] = [⟨A, B⟩] → ((z 𝐺 w 𝐺) ↔ ⟨A, B (𝐺 × 𝐺)))
3225, 31syl5ibr 145 . . . . . . 7 ([⟨z, w⟩] = [⟨A, B⟩] → ((A 𝐺 B 𝐺) → (z 𝐺 w 𝐺)))
33 opelxpi 4303 . . . . . . . 8 ((𝐶 𝐺 𝐷 𝐺) → ⟨𝐶, 𝐷 (𝐺 × 𝐺))
34 opelxp 4301 . . . . . . . . 9 (⟨v, u (𝐺 × 𝐺) ↔ (v 𝐺 u 𝐺))
3527a1i 9 . . . . . . . . . 10 ([⟨v, u⟩] = [⟨𝐶, 𝐷⟩] Er (𝐺 × 𝐺))
36 id 19 . . . . . . . . . 10 ([⟨v, u⟩] = [⟨𝐶, 𝐷⟩] → [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] )
3735, 36ereldm 6060 . . . . . . . . 9 ([⟨v, u⟩] = [⟨𝐶, 𝐷⟩] → (⟨v, u (𝐺 × 𝐺) ↔ ⟨𝐶, 𝐷 (𝐺 × 𝐺)))
3834, 37syl5bbr 183 . . . . . . . 8 ([⟨v, u⟩] = [⟨𝐶, 𝐷⟩] → ((v 𝐺 u 𝐺) ↔ ⟨𝐶, 𝐷 (𝐺 × 𝐺)))
3933, 38syl5ibr 145 . . . . . . 7 ([⟨v, u⟩] = [⟨𝐶, 𝐷⟩] → ((𝐶 𝐺 𝐷 𝐺) → (v 𝐺 u 𝐺)))
4032, 39im2anan9 517 . . . . . 6 (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) → (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → ((z 𝐺 w 𝐺) (v 𝐺 u 𝐺))))
41 brecop.6 . . . . . . . . 9 ((((z 𝐺 w 𝐺) (A 𝐺 B 𝐺)) ((v 𝐺 u 𝐺) (𝐶 𝐺 𝐷 𝐺))) → (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) → (φψ)))
4241an4s 509 . . . . . . . 8 ((((z 𝐺 w 𝐺) (v 𝐺 u 𝐺)) ((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺))) → (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) → (φψ)))
4342ex 108 . . . . . . 7 (((z 𝐺 w 𝐺) (v 𝐺 u 𝐺)) → (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) → (φψ))))
4443com13 74 . . . . . 6 (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) → (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → (((z 𝐺 w 𝐺) (v 𝐺 u 𝐺)) → (φψ))))
4540, 44mpdd 36 . . . . 5 (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) → (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → (φψ)))
4645pm5.74d 171 . . . 4 (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) → ((((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → φ) ↔ (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → ψ)))
4724, 46cgsex4g 2568 . . 3 (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → (zwvu(([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → φ)) ↔ (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → ψ)))
48 eqcom 2024 . . . . . . 7 ([⟨A, B⟩] = [⟨z, w⟩] ↔ [⟨z, w⟩] = [⟨A, B⟩] )
49 eqcom 2024 . . . . . . 7 ([⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ↔ [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] )
5048, 49anbi12i 436 . . . . . 6 (([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) ↔ ([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ))
5150a1i 9 . . . . 5 (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → (([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) ↔ ([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] )))
52 biimt 230 . . . . 5 (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → (φ ↔ (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → φ)))
5351, 52anbi12d 445 . . . 4 (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → ((([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) φ) ↔ (([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → φ))))
54534exbidv 1732 . . 3 (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → (zwvu(([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) φ) ↔ zwvu(([⟨z, w⟩] = [⟨A, B⟩] [⟨v, u⟩] = [⟨𝐶, 𝐷⟩] ) (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → φ))))
55 biimt 230 . . 3 (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → (ψ ↔ (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → ψ)))
5647, 54, 553bitr4d 209 . 2 (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → (zwvu(([⟨A, B⟩] = [⟨z, w⟩] [⟨𝐶, 𝐷⟩] = [⟨v, u⟩] ) φ) ↔ ψ))
5719, 56bitrd 177 1 (((A 𝐺 B 𝐺) (𝐶 𝐺 𝐷 𝐺)) → ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535  cop 3353   class class class wbr 3738  {copab 3791   × cxp 4270   Er wer 6014  [cec 6015   / cqs 6016
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-er 6017  df-ec 6019  df-qs 6023
This theorem is referenced by:  ordpipqqs  6233  ltsrprg  6494
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