Theorem brecop2 3243

Description: Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis.

Hypotheses

Ref	Expression
brecop2.1	|- S e. V
brecop2.2	|- B e. V
brecop2.3	|- C e. V
brecop2.4	|- D e. V
brecop2.5	|- Er S
brecop2.6	|- dom S = (G X. G)
brecop2.7	|- H = ((G X. G)/.S)
brecop2.8	|- R (_ (H X. H)
brecop2.9	|- Q (_ (G X. G)
brecop2.10	|- -. (/) e. G
brecop2.11	|- dom F = (G X. G)
brecop2.12	|- (((A e. G /\ B e. G) /\ (C e. G /\ D e. G)) -> ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC)))

Assertion

Ref	Expression
brecop2	|- ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC))

Proof of Theorem brecop2

Step	Hyp	Ref	Expression
1		brecop2.1	. . . . 5 |- S e. V
2		ecexg 3204	. . . . 5 |- (S e. V -> [<.C, D>.]S e. V)
3	1, 2	ax-mp 6	. . . 4 |- [<.C, D>.]S e. V
4		brecop2.8	. . . 4 |- R (_ (H X. H)
5	3, 4	brel 2459	. . 3 |- ([<.A, B>.]SR[<.C, D>.]S -> ([<.A, B>.]S e. H /\ [<.C, D>.]S e. H))
6		brecop2.7	. . . . . . 7 |- H = ((G X. G)/.S)
7	6	eleq2i 1153	. . . . . 6 |- ([<.A, B>.]S e. H <-> [<.A, B>.]S e. ((G X. G)/.S))
8		opex 1893	. . . . . . 7 |- <.A, B>. e. V
9		brecop2.5	. . . . . . 7 |- Er S
10		brecop2.6	. . . . . . 7 |- dom S = (G X. G)
11	8, 9, 10	ecelqsdm 3235	. . . . . 6 |- ([<.A, B>.]S e. ((G X. G)/.S) -> <.A, B>. e. (G X. G))
12	7, 11	sylbi 174	. . . . 5 |- ([<.A, B>.]S e. H -> <.A, B>. e. (G X. G))
13		brecop2.2	. . . . . 6 |- B e. V
14	13	opelxp 2452	. . . . 5 |- (<.A, B>. e. (G X. G) <-> (A e. G /\ B e. G))
15	12, 14	sylib 173	. . . 4 |- ([<.A, B>.]S e. H -> (A e. G /\ B e. G))
16	6	eleq2i 1153	. . . . . 6 |- ([<.C, D>.]S e. H <-> [<.C, D>.]S e. ((G X. G)/.S))
17		opex 1893	. . . . . . 7 |- <.C, D>. e. V
18	17, 9, 10	ecelqsdm 3235	. . . . . 6 |- ([<.C, D>.]S e. ((G X. G)/.S) -> <.C, D>. e. (G X. G))
19	16, 18	sylbi 174	. . . . 5 |- ([<.C, D>.]S e. H -> <.C, D>. e. (G X. G))
20		brecop2.4	. . . . . 6 |- D e. V
21	20	opelxp 2452	. . . . 5 |- (<.C, D>. e. (G X. G) <-> (C e. G /\ D e. G))
22	19, 21	sylib 173	. . . 4 |- ([<.C, D>.]S e. H -> (C e. G /\ D e. G))
23	15, 22	anim12i 268	. . 3 |- (([<.A, B>.]S e. H /\ [<.C, D>.]S e. H) -> ((A e. G /\ B e. G) /\ (C e. G /\ D e. G)))
24	5, 23	syl 12	. 2 |- ([<.A, B>.]SR[<.C, D>.]S -> ((A e. G /\ B e. G) /\ (C e. G /\ D e. G)))
25		oprex 3018	. . . . 5 |- (BFC) e. V
26		brecop2.9	. . . . 5 |- Q (_ (G X. G)
27	25, 26	brel 2459	. . . 4 |- ((AFD)Q(BFC) -> ((AFD) e. G /\ (BFC) e. G))
28		brecop2.11	. . . . . 6 |- dom F = (G X. G)
29		brecop2.10	. . . . . 6 |- -. (/) e. G
30	20, 28, 29	ndmoprrcl 3060	. . . . 5 |- ((AFD) e. G -> (A e. G /\ D e. G))
31		brecop2.3	. . . . . 6 |- C e. V
32	31, 28, 29	ndmoprrcl 3060	. . . . 5 |- ((BFC) e. G -> (B e. G /\ C e. G))
33	30, 32	anim12i 268	. . . 4 |- (((AFD) e. G /\ (BFC) e. G) -> ((A e. G /\ D e. G) /\ (B e. G /\ C e. G)))
34	27, 33	syl 12	. . 3 |- ((AFD)Q(BFC) -> ((A e. G /\ D e. G) /\ (B e. G /\ C e. G)))
35		an42 389	. . 3 |- (((A e. G /\ D e. G) /\ (B e. G /\ C e. G)) <-> ((A e. G /\ B e. G) /\ (C e. G /\ D e. G)))
36	34, 35	sylib 173	. 2 |- ((AFD)Q(BFC) -> ((A e. G /\ B e. G) /\ (C e. G /\ D e. G)))
37		brecop2.12	. 2 |- (((A e. G /\ B e. G) /\ (C e. G /\ D e. G)) -> ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC)))
38	24, 36, 37	pm5.21nii 504	1 |- ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  (/)c0 1707  <.cop 1810   class class class wbr 2054   X. cxp 2408  dom cdm 2410  (class class class)co 3001  Er wer 3197  [cec 3198  /.cqs 3199

This theorem is referenced by:  ordpipq 3850  ltsrpr 3980

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  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  df-ec 3202  df-qs 3205

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