Theorem ecopopreq 3244

Description: Express the relation R (specified by the hypothesis) in terms of its operation F.

Hypothesis

Ref	Expression
ecopopr.1	⊢ R = {⟨x, y⟩∣((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (zFu) = (wFv)))}

Assertion

Ref	Expression
ecopopreq	⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (⟨A, B⟩R⟨C, D⟩ ↔ (AFD) = (BFC)))

Distinct variable group(s):   x,y,z,w,v,u,F   x,S,y,z,w,v,u   x,A,y,z,w,v,u   x,B,y,z,w,v,u   x,C,y,z,w,v,u   x,D,y,z,w,v,u

Proof of Theorem ecopopreq

Step	Hyp	Ref	Expression
1		opreq12 3008	. . . 4 ⊢ ((z = A ∧ u = D) → (zFu) = (AFD))
2		opreq12 3008	. . . 4 ⊢ ((w = B ∧ v = C) → (wFv) = (BFC))
3	1, 2	cleqan12d 1116	. . 3 ⊢ (((z = A ∧ u = D) ∧ (w = B ∧ v = C)) → ((zFu) = (wFv) ↔ (AFD) = (BFC)))
4	3	an42s 391	. 2 ⊢ (((z = A ∧ w = B) ∧ (v = C ∧ u = D)) → ((zFu) = (wFv) ↔ (AFD) = (BFC)))
5		ecopopr.1	. 2 ⊢ R = {⟨x, y⟩∣((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (zFu) = (wFv)))}
6	4, 5	opbrop 2472	1 ⊢ (((A ∈ S ∧ B ∈ S) ∧ (C ∈ S ∧ D ∈ S)) → (⟨A, B⟩R⟨C, D⟩ ↔ (AFD) = (BFC)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  {copab 2055   × cxp 2408  (class class class)co 3001

This theorem is referenced by:  ecopoprdm 3245  ecopoprsym 3246  ecopoprtrn 3247  enqbreq 3838  enrbreq 3968

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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