Theorem ecopoprsym 3246

Description: Assuming the operation F is commutative, show that the relation R, specified by the first hypothesis, is symmetric.

Hypotheses

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)))}
ecopopr.com	⊢ (xFy) = (yFx)
ecopopr.2	⊢ B ∈ V

Assertion

Ref	Expression
ecopoprsym	⊢ (ARB → BRA)

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

Proof of Theorem ecopoprsym

Step	Hyp	Ref	Expression
1		ecopopr.2	. . . 4 ⊢ B ∈ V
2		ecopopr.1	. . . . 5 ⊢ R = {⟨x, y⟩∣((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (zFu) = (wFv)))}
3		opabssxp 2468	. . . . 5 ⊢ {⟨x, y⟩∣((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (zFu) = (wFv)))} ⊆ ((S × S) × (S × S))
4	2, 3	eqsstr 1530	. . . 4 ⊢ R ⊆ ((S × S) × (S × S))
5	1, 4	brel 2459	. . 3 ⊢ (ARB → (A ∈ (S × S) ∧ B ∈ (S × S)))
6		cleqid 1102	. . . 4 ⊢ (S × S) = (S × S)
7		breq1 2065	. . . . 5 ⊢ (⟨f, g⟩ = A → (⟨f, g⟩R⟨h, t⟩ ↔ AR⟨h, t⟩))
8		breq2 2066	. . . . 5 ⊢ (⟨f, g⟩ = A → (⟨h, t⟩R⟨f, g⟩ ↔ ⟨h, t⟩RA))
9	7, 8	bibi12d 477	. . . 4 ⊢ (⟨f, g⟩ = A → ((⟨f, g⟩R⟨h, t⟩ ↔ ⟨h, t⟩R⟨f, g⟩) ↔ (AR⟨h, t⟩ ↔ ⟨h, t⟩RA)))
10		breq2 2066	. . . . 5 ⊢ (⟨h, t⟩ = B → (AR⟨h, t⟩ ↔ ARB))
11		breq1 2065	. . . . 5 ⊢ (⟨h, t⟩ = B → (⟨h, t⟩RA ↔ BRA))
12	10, 11	bibi12d 477	. . . 4 ⊢ (⟨h, t⟩ = B → ((AR⟨h, t⟩ ↔ ⟨h, t⟩RA) ↔ (ARB ↔ BRA)))
13	2	ecopopreq 3244	. . . . . 6 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S)) → (⟨f, g⟩R⟨h, t⟩ ↔ (fFt) = (gFh)))
14		visset 1350	. . . . . . . . 9 ⊢ f ∈ V
15		visset 1350	. . . . . . . . 9 ⊢ t ∈ V
16		ecopopr.com	. . . . . . . . 9 ⊢ (xFy) = (yFx)
17	14, 15, 16	caoprcom 3067	. . . . . . . 8 ⊢ (fFt) = (tFf)
18		visset 1350	. . . . . . . . 9 ⊢ g ∈ V
19		visset 1350	. . . . . . . . 9 ⊢ h ∈ V
20	18, 19, 16	caoprcom 3067	. . . . . . . 8 ⊢ (gFh) = (hFg)
21	17, 20	cleq12i 1114	. . . . . . 7 ⊢ ((fFt) = (gFh) ↔ (tFf) = (hFg))
22		cleqcom 1103	. . . . . . 7 ⊢ ((tFf) = (hFg) ↔ (hFg) = (tFf))
23	21, 22	bitr 151	. . . . . 6 ⊢ ((fFt) = (gFh) ↔ (hFg) = (tFf))
24	13, 23	syl6bb 414	. . . . 5 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S)) → (⟨f, g⟩R⟨h, t⟩ ↔ (hFg) = (tFf)))
25	2	ecopopreq 3244	. . . . . 6 ⊢ (((h ∈ S ∧ t ∈ S) ∧ (f ∈ S ∧ g ∈ S)) → (⟨h, t⟩R⟨f, g⟩ ↔ (hFg) = (tFf)))
26	25	ancoms 334	. . . . 5 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S)) → (⟨h, t⟩R⟨f, g⟩ ↔ (hFg) = (tFf)))
27	24, 26	bitr4d 409	. . . 4 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S)) → (⟨f, g⟩R⟨h, t⟩ ↔ ⟨h, t⟩R⟨f, g⟩))
28	6, 9, 12, 27	2optocl 2470	. . 3 ⊢ ((A ∈ (S × S) ∧ B ∈ (S × S)) → (ARB ↔ BRA))
29	5, 28	syl 12	. 2 ⊢ (ARB → (ARB ↔ BRA))
30	29	ibi 449	1 ⊢ (ARB → BRA)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  {copab 2055   × cxp 2408  (class class class)co 3001

This theorem is referenced by:  ecopoprer 3248

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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