Theorem spwpr4 10006

Description: Supremum of an unordered pair.

Hypothesis

Ref	Expression
spwpr4.1	|- X = dom R

Assertion

Ref	Expression
spwpr4	|- (((R e. Poset /\ C e. D) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (R supw {A, B}) = C)

Distinct variable groups:   x,A   x,B   x,C   x,R   x,X

Proof of Theorem spwpr4

Step	Hyp	Ref	Expression
1		simp1l 900	. . . 4 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> R e. Poset)
2		prex 3526	. . . . 5 |- {A, B} e. _V
3	2	a1i 8	. . . 4 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> {A, B} e. _V)
4		breq2 3342	. . . . . . . . . 10 |- (y = C -> (ARy <-> ARC))
5		breq2 3342	. . . . . . . . . 10 |- (y = C -> (BRy <-> BRC))
6	4, 5	anbi12d 690	. . . . . . . . 9 |- (y = C -> ((ARy /\ BRy) <-> (ARC /\ BRC)))
7		breq1 3341	. . . . . . . . . . 11 |- (y = C -> (yRx <-> CRx))
8	7	imbi2d 674	. . . . . . . . . 10 |- (y = C -> (((ARx /\ BRx) -> yRx) <-> ((ARx /\ BRx) -> CRx)))
9	8	ralbidv 2123	. . . . . . . . 9 |- (y = C -> (A.x e. X ((ARx /\ BRx) -> yRx) <-> A.x e. X ((ARx /\ BRx) -> CRx)))
10	6, 9	anbi12d 690	. . . . . . . 8 |- (y = C -> (((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx)) <-> ((ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx))))
11	10	rcla4ev 2381	. . . . . . 7 |- ((C e. X /\ ((ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx))) -> E.y e. X ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx)))
12	11	3impb 1063	. . . . . 6 |- ((C e. X /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> E.y e. X ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx)))
13	12	3adant1l 1090	. . . . 5 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> E.y e. X ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx)))
14		simpll 448	. . . . . . . 8 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC)) -> R e. Poset)
15		psrel 9989	. . . . . . . . . . 11 |- (R e. Poset -> Rel R)
16		brrelex 4028	. . . . . . . . . . . . 13 |- ((Rel R /\ ARC) -> A e. _V)
17	16	ex 402	. . . . . . . . . . . 12 |- (Rel R -> (ARC -> A e. _V))
18		brrelex 4028	. . . . . . . . . . . . 13 |- ((Rel R /\ BRC) -> B e. _V)
19	18	ex 402	. . . . . . . . . . . 12 |- (Rel R -> (BRC -> B e. _V))
20	17, 19	anim12d 617	. . . . . . . . . . 11 |- (Rel R -> ((ARC /\ BRC) -> (A e. _V /\ B e. _V)))
21	15, 20	syl 12	. . . . . . . . . 10 |- (R e. Poset -> ((ARC /\ BRC) -> (A e. _V /\ B e. _V)))
22	21	imp 377	. . . . . . . . 9 |- ((R e. Poset /\ (ARC /\ BRC)) -> (A e. _V /\ B e. _V))
23	22	adantlr 429	. . . . . . . 8 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC)) -> (A e. _V /\ B e. _V))
24		eqid 1884	. . . . . . . . 9 |- {A, B} = {A, B}
25		biid 187	. . . . . . . . . 10 |- ((A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)) <-> (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)))
26	25	spwpr2 10001	. . . . . . . . 9 |- (((R e. Poset /\ {A, B} = {A, B}) /\ (A e. _V /\ B e. _V)) -> ((A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)) <-> ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))))
27	24, 26	mpanl2 771	. . . . . . . 8 |- ((R e. Poset /\ (A e. _V /\ B e. _V)) -> ((A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)) <-> ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))))
28	14, 23, 27	syl11anc 524	. . . . . . 7 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC)) -> ((A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)) <-> ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))))
29	28	rexbidv 2124	. . . . . 6 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC)) -> (E.y e. X (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)) <-> E.y e. X ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))))
30	29	3adant3 896	. . . . 5 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (E.y e. X (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)) <-> E.y e. X ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))))
31	13, 30	mpbird 213	. . . 4 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> E.y e. X (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)))
32		spwpr4.1	. . . . 5 |- X = dom R
33	32, 25	spwval 10002	. . . 4 |- ((R e. Poset /\ {A, B} e. _V /\ E.y e. X (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx))) -> (R supw {A, B}) = U.{y e. X | (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx))})
34	1, 3, 31, 33	syl111anc 1100	. . 3 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (R supw {A, B}) = U.{y e. X | (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx))})
35	28	rabbidv 2287	. . . . 5 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC)) -> {y e. X | (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx))} = {y e. X | ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))})
36	35	unieqd 3188	. . . 4 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC)) -> U.{y e. X | (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx))} = U.{y e. X | ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))})
37	36	3adant3 896	. . 3 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> U.{y e. X | (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx))} = U.{y e. X | ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))})
38		3simpc 874	. . . 4 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> ((ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)))
39		simp1r 901	. . . . 5 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> C e. X)
40	25	spweu 10000	. . . . . . 7 |- ((R e. Poset /\ E.y e. X (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx))) -> E!y e. X (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)))
41	1, 31, 40	syl11anc 524	. . . . . 6 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> E!y e. X (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)))
42	28	reubidv 2260	. . . . . . 7 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC)) -> (E!y e. X (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)) <-> E!y e. X ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))))
43	42	3adant3 896	. . . . . 6 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (E!y e. X (A.x e. {A, B}xRy /\ A.x e. X (A.z e. {A, B}zRx -> yRx)) <-> E!y e. X ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))))
44	41, 43	mpbid 212	. . . . 5 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> E!y e. X ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx)))
45	10	reuuni2 3811	. . . . 5 |- ((C e. X /\ E!y e. X ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))) -> (((ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) <-> U.{y e. X | ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))} = C))
46	39, 44, 45	syl11anc 524	. . . 4 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (((ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) <-> U.{y e. X | ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))} = C))
47	38, 46	mpbid 212	. . 3 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> U.{y e. X | ((ARy /\ BRy) /\ A.x e. X ((ARx /\ BRx) -> yRx))} = C)
48	34, 37, 47	3eqtrd 1929	. 2 |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (R supw {A, B}) = C)
49		simpll 448	. . . 4 |- (((R e. Poset /\ C e. D) /\ (ARC /\ BRC)) -> R e. Poset)
50		relelrng 4194	. . . . . . . . 9 |- ((C e. D /\ Rel R /\ ARC) -> C e. ran R)
51	50, 15	syl3an2 1131	. . . . . . . 8 |- ((C e. D /\ R e. Poset /\ ARC) -> C e. ran R)
52	51	3com12 1071	. . . . . . 7 |- ((R e. Poset /\ C e. D /\ ARC) -> C e. ran R)
53	52	3expa 1067	. . . . . 6 |- (((R e. Poset /\ C e. D) /\ ARC) -> C e. ran R)
54	32	psrn 9993	. . . . . . 7 |- (R e. Poset -> X = ran R)
55	54	ad2antrr 440	. . . . . 6 |- (((R e. Poset /\ C e. D) /\ ARC) -> X = ran R)
56	53, 55	eleqtrrd 1974	. . . . 5 |- (((R e. Poset /\ C e. D) /\ ARC) -> C e. X)
57	56	adantrr 431	. . . 4 |- (((R e. Poset /\ C e. D) /\ (ARC /\ BRC)) -> C e. X)
58	49, 57	jca 310	. . 3 |- (((R e. Poset /\ C e. D) /\ (ARC /\ BRC)) -> (R e. Poset /\ C e. X))
59	58	3adant3 896	. 2 |- (((R e. Poset /\ C e. D) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (R e. Poset /\ C e. X))
60	48, 59	syld3an1 1143	1 |- (((R e. Poset /\ C e. D) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (R supw {A, B}) = C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  {crab 2108  _Vcvv 2292  {cpr 3045  U.cuni 3177   class class class wbr 3338  dom cdm 3986  ran crn 3987  Rel wrel 3991  (class class class)co 4884  Posetcps 9980   sup_w cspw 9981

This theorem is referenced by:  spwpr4c 10009  toplat 14638

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-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-ps 9984  df-spw 9985

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