Theorem brecop2 5366

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	|- dom S = (G X. G)
brecop2.6	|- H = ((G X. G)/.S)
brecop2.7	|- R C_ (H X. H)
brecop2.8	|- Q C_ (G X. G)
brecop2.9	|- -. (/) e. G
brecop2.10	|- dom F = (G X. G)
brecop2.11	|- (((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 5322	. . . . 5 |- (S e. _V -> [<.C, D>.]S e. _V)
3	1, 2	ax-mp 7	. . . 4 |- [<.C, D>.]S e. _V
4		brecop2.7	. . . 4 |- R C_ (H X. H)
5	3, 4	brel 4048	. . 3 |- ([<.A, B>.]SR[<.C, D>.]S -> ([<.A, B>.]S e. H /\ [<.C, D>.]S e. H))
6		brecop2.6	. . . . . . 7 |- H = ((G X. G)/.S)
7	6	eleq2i 1961	. . . . . 6 |- ([<.A, B>.]S e. H <-> [<.A, B>.]S e. ((G X. G)/.S))
8		opex 3527	. . . . . . 7 |- <.A, B>. e. _V
9		brecop2.5	. . . . . . 7 |- dom S = (G X. G)
10	8, 9	ecelqsdm 5358	. . . . . 6 |- ([<.A, B>.]S e. ((G X. G)/.S) -> <.A, B>. e. (G X. G))
11	7, 10	sylbi 216	. . . . 5 |- ([<.A, B>.]S e. H -> <.A, B>. e. (G X. G))
12		brecop2.2	. . . . . 6 |- B e. _V
13	12	opelxp 4036	. . . . 5 |- (<.A, B>. e. (G X. G) <-> (A e. G /\ B e. G))
14	11, 13	sylib 215	. . . 4 |- ([<.A, B>.]S e. H -> (A e. G /\ B e. G))
15	6	eleq2i 1961	. . . . . 6 |- ([<.C, D>.]S e. H <-> [<.C, D>.]S e. ((G X. G)/.S))
16		opex 3527	. . . . . . 7 |- <.C, D>. e. _V
17	16, 9	ecelqsdm 5358	. . . . . 6 |- ([<.C, D>.]S e. ((G X. G)/.S) -> <.C, D>. e. (G X. G))
18	15, 17	sylbi 216	. . . . 5 |- ([<.C, D>.]S e. H -> <.C, D>. e. (G X. G))
19		brecop2.4	. . . . . 6 |- D e. _V
20	19	opelxp 4036	. . . . 5 |- (<.C, D>. e. (G X. G) <-> (C e. G /\ D e. G))
21	18, 20	sylib 215	. . . 4 |- ([<.C, D>.]S e. H -> (C e. G /\ D e. G))
22	14, 21	anim12i 360	. . 3 |- (([<.A, B>.]S e. H /\ [<.C, D>.]S e. H) -> ((A e. G /\ B e. G) /\ (C e. G /\ D e. G)))
23	5, 22	syl 12	. 2 |- ([<.A, B>.]SR[<.C, D>.]S -> ((A e. G /\ B e. G) /\ (C e. G /\ D e. G)))
24		oprex 4907	. . . . 5 |- (BFC) e. _V
25		brecop2.8	. . . . 5 |- Q C_ (G X. G)
26	24, 25	brel 4048	. . . 4 |- ((AFD)Q(BFC) -> ((AFD) e. G /\ (BFC) e. G))
27		brecop2.10	. . . . . 6 |- dom F = (G X. G)
28		brecop2.9	. . . . . 6 |- -. (/) e. G
29	19, 27, 28	ndmoprrcl 4979	. . . . 5 |- ((AFD) e. G -> (A e. G /\ D e. G))
30		brecop2.3	. . . . . 6 |- C e. _V
31	30, 27, 28	ndmoprrcl 4979	. . . . 5 |- ((BFC) e. G -> (B e. G /\ C e. G))
32	29, 31	anim12i 360	. . . 4 |- (((AFD) e. G /\ (BFC) e. G) -> ((A e. G /\ D e. G) /\ (B e. G /\ C e. G)))
33	26, 32	syl 12	. . 3 |- ((AFD)Q(BFC) -> ((A e. G /\ D e. G) /\ (B e. G /\ C e. G)))
34		an42 565	. . 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)))
35	33, 34	sylib 215	. 2 |- ((AFD)Q(BFC) -> ((A e. G /\ B e. G) /\ (C e. G /\ D e. G)))
36		brecop2.11	. 2 |- (((A e. G /\ B e. G) /\ (C e. G /\ D e. G)) -> ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC)))
37	23, 35, 36	pm5.21nii 743	1 |- ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  (/)c0 2875  <.cop 3046   class class class wbr 3338   X. cxp 3984  dom cdm 3986  (class class class)co 4884  [cec 5316  /.cqs 5317

This theorem is referenced by:  ordpipq 6208  ltsrpr 6338

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-ec 5320  df-qs 5323

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