Theorem brecop2 4307

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 (_ (H X. H)
brecop2.8	|- Q (_ (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 4265	. . . . 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 (_ (H X. H)
5	3, 4	brel 3223	. . 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 1538	. . . . . 6 |- ([<.A, B>.]S e. H <-> [<.A, B>.]S e. ((G X. G)/.S))
8		opex 2782	. . . . . . 7 |- <.A, B>. e. V
9		brecop2.5	. . . . . . 7 |- dom S = (G X. G)
10	8, 9	ecelqsdm 4299	. . . . . 6 |- ([<.A, B>.]S e. ((G X. G)/.S) -> <.A, B>. e. (G X. G))
11	7, 10	sylbi 199	. . . . 5 |- ([<.A, B>.]S e. H -> <.A, B>. e. (G X. G))
12		brecop2.2	. . . . . 6 |- B e. V
13	12	opelxp 3214	. . . . 5 |- (<.A, B>. e. (G X. G) <-> (A e. G /\ B e. G))
14	11, 13	sylib 198	. . . 4 |- ([<.A, B>.]S e. H -> (A e. G /\ B e. G))
15	6	eleq2i 1538	. . . . . 6 |- ([<.C, D>.]S e. H <-> [<.C, D>.]S e. ((G X. G)/.S))
16		opex 2782	. . . . . . 7 |- <.C, D>. e. V
17	16, 9	ecelqsdm 4299	. . . . . 6 |- ([<.C, D>.]S e. ((G X. G)/.S) -> <.C, D>. e. (G X. G))
18	15, 17	sylbi 199	. . . . 5 |- ([<.C, D>.]S e. H -> <.C, D>. e. (G X. G))
19		brecop2.4	. . . . . 6 |- D e. V
20	19	opelxp 3214	. . . . 5 |- (<.C, D>. e. (G X. G) <-> (C e. G /\ D e. G))
21	18, 20	sylib 198	. . . 4 |- ([<.C, D>.]S e. H -> (C e. G /\ D e. G))
22	14, 21	anim12i 333	. . 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 10	. 2 |- ([<.A, B>.]SR[<.C, D>.]S -> ((A e. G /\ B e. G) /\ (C e. G /\ D e. G)))
24		oprex 3983	. . . . 5 |- (BFC) e. V
25		brecop2.8	. . . . 5 |- Q (_ (G X. G)
26	24, 25	brel 3223	. . . 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 4046	. . . . 5 |- ((AFD) e. G -> (A e. G /\ D e. G))
30		brecop2.3	. . . . . 6 |- C e. V
31	30, 27, 28	ndmoprrcl 4046	. . . . 5 |- ((BFC) e. G -> (B e. G /\ C e. G))
32	29, 31	anim12i 333	. . . 4 |- (((AFD) e. G /\ (BFC) e. G) -> ((A e. G /\ D e. G) /\ (B e. G /\ C e. G)))
33	26, 32	syl 10	. . 3 |- ((AFD)Q(BFC) -> ((A e. G /\ D e. G) /\ (B e. G /\ C e. G)))
34		an42 507	. . 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 198	. 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 679	1 |- ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   (_ wss 2047  (/)c0 2280  <.cop 2411   class class class wbr 2619   X. cxp 3168  dom cdm 3170  (class class class)co 3963  [cec 4259  /.cqs 4260

This theorem is referenced by:  ordpipq 5056  ltsrpr 5186

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-ec 4263  df-qs 4266

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