Theorem dfps2 14634

Description: Alternate definition of a poset. Bourbaki E.III.2 prop. 1.

Assertion

Ref	Expression
dfps2	|- Poset = {r | (Rel r /\ (r o. r) = r /\ (r i^i `'r) = ( _I |` U.U.r))}

Proof of Theorem dfps2

Step	Hyp	Ref	Expression
1		df-ps 9984	. 2 |- Poset = {r | (Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r))}
2		simp1 876	. . . . 5 |- ((Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r)) -> Rel r)
3		eqimss2 2667	. . . . . . . 8 |- ((r i^i `'r) = ( _I |` U.U.r) -> ( _I |` U.U.r) C_ (r i^i `'r))
4		ssin 2814	. . . . . . . . 9 |- ((( _I |` U.U.r) C_ r /\ ( _I |` U.U.r) C_ `'r) <-> ( _I |` U.U.r) C_ (r i^i `'r))
5		ref4w 14370	. . . . . . . . . . 11 |- (A.x e. U.U.rxrx <-> ( _I |` U.U.r) C_ r)
6		domrngref 14364	. . . . . . . . . . . . . . 15 |- ((Rel r /\ A.x e. U.U.rxrx) -> dom r = ran r)
7		domfldref 14365	. . . . . . . . . . . . . . 15 |- ((Rel r /\ A.x e. U.U.rxrx) -> dom r = U.U.r)
8		eqtr 1904	. . . . . . . . . . . . . . . . . 18 |- ((ran r = dom r /\ dom r = U.U.r) -> ran r = U.U.r)
9		reseq2 4219	. . . . . . . . . . . . . . . . . . . . . 22 |- (U.U.r = ran r -> ( _I |` U.U.r) = ( _I |` ran r))
10	9	eqcoms 1887	. . . . . . . . . . . . . . . . . . . . 21 |- (ran r = U.U.r -> ( _I |` U.U.r) = ( _I |` ran r))
11	10	eqeq2d 1895	. . . . . . . . . . . . . . . . . . . 20 |- (ran r = U.U.r -> ((r i^i `'r) = ( _I |` U.U.r) <-> (r i^i `'r) = ( _I |` ran r)))
12		inss1 2812	. . . . . . . . . . . . . . . . . . . . 21 |- (r i^i `'r) C_ r
13		ssid 2634	. . . . . . . . . . . . . . . . . . . . . 22 |- ran r C_ ran r
14		cores 4400	. . . . . . . . . . . . . . . . . . . . . 22 |- (ran r C_ ran r -> (( _I |` ran r) o. r) = ( _I o. r))
15	13, 14	ax-mp 7	. . . . . . . . . . . . . . . . . . . . 21 |- (( _I |` ran r) o. r) = ( _I o. r)
16		sseq1 2637	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((r i^i `'r) = ( _I |` ran r) -> ((r i^i `'r) C_ r <-> ( _I |` ran r) C_ r))
17		inclrel 14444	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (( _I |` ran r) C_ r -> (( _I |` ran r) o. r) C_ (r o. r))
18		coi2 4414	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (Rel r -> ( _I o. r) = r)
19		eqtr 1904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (((( _I |` ran r) o. r) = ( _I o. r) /\ ( _I o. r) = r) -> (( _I |` ran r) o. r) = r)
20		sseq1 2637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((( _I |` ran r) o. r) = r -> ((( _I |` ran r) o. r) C_ (r o. r) <-> r C_ (r o. r)))
21		eqss 2631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((r o. r) = r <-> ((r o. r) C_ r /\ r C_ (r o. r)))
22	21	biimpri 169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (((r o. r) C_ r /\ r C_ (r o. r)) -> (r o. r) = r)
23	22	expcom 403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (r C_ (r o. r) -> ((r o. r) C_ r -> (r o. r) = r))
24	20, 23	syl6bi 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((( _I |` ran r) o. r) = r -> ((( _I |` ran r) o. r) C_ (r o. r) -> ((r o. r) C_ r -> (r o. r) = r)))
25	19, 24	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (((( _I |` ran r) o. r) = ( _I o. r) /\ ( _I o. r) = r) -> ((( _I |` ran r) o. r) C_ (r o. r) -> ((r o. r) C_ r -> (r o. r) = r)))
26	25	ex 402	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((( _I |` ran r) o. r) = ( _I o. r) -> (( _I o. r) = r -> ((( _I |` ran r) o. r) C_ (r o. r) -> ((r o. r) C_ r -> (r o. r) = r))))
27	26	com3l 38	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (( _I o. r) = r -> ((( _I |` ran r) o. r) C_ (r o. r) -> ((( _I |` ran r) o. r) = ( _I o. r) -> ((r o. r) C_ r -> (r o. r) = r))))
28	18, 27	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (Rel r -> ((( _I |` ran r) o. r) C_ (r o. r) -> ((( _I |` ran r) o. r) = ( _I o. r) -> ((r o. r) C_ r -> (r o. r) = r))))
29	28	com3l 38	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((( _I |` ran r) o. r) C_ (r o. r) -> ((( _I |` ran r) o. r) = ( _I o. r) -> (Rel r -> ((r o. r) C_ r -> (r o. r) = r))))
30	17, 29	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- (( _I |` ran r) C_ r -> ((( _I |` ran r) o. r) = ( _I o. r) -> (Rel r -> ((r o. r) C_ r -> (r o. r) = r))))
31	16, 30	syl6bi 231	. . . . . . . . . . . . . . . . . . . . . 22 |- ((r i^i `'r) = ( _I |` ran r) -> ((r i^i `'r) C_ r -> ((( _I |` ran r) o. r) = ( _I o. r) -> (Rel r -> ((r o. r) C_ r -> (r o. r) = r)))))
32	31	com3l 38	. . . . . . . . . . . . . . . . . . . . 21 |- ((r i^i `'r) C_ r -> ((( _I |` ran r) o. r) = ( _I o. r) -> ((r i^i `'r) = ( _I |` ran r) -> (Rel r -> ((r o. r) C_ r -> (r o. r) = r)))))
33	12, 15, 32	mp2 54	. . . . . . . . . . . . . . . . . . . 20 |- ((r i^i `'r) = ( _I |` ran r) -> (Rel r -> ((r o. r) C_ r -> (r o. r) = r)))
34	11, 33	syl6bi 231	. . . . . . . . . . . . . . . . . . 19 |- (ran r = U.U.r -> ((r i^i `'r) = ( _I |` U.U.r) -> (Rel r -> ((r o. r) C_ r -> (r o. r) = r))))
35	34	com23 36	. . . . . . . . . . . . . . . . . 18 |- (ran r = U.U.r -> (Rel r -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (r o. r) = r))))
36	8, 35	syl 12	. . . . . . . . . . . . . . . . 17 |- ((ran r = dom r /\ dom r = U.U.r) -> (Rel r -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (r o. r) = r))))
37	36	ex 402	. . . . . . . . . . . . . . . 16 |- (ran r = dom r -> (dom r = U.U.r -> (Rel r -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (r o. r) = r)))))
38	37	eqcoms 1887	. . . . . . . . . . . . . . 15 |- (dom r = ran r -> (dom r = U.U.r -> (Rel r -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (r o. r) = r)))))
39	6, 7, 38	sylc 83	. . . . . . . . . . . . . 14 |- ((Rel r /\ A.x e. U.U.rxrx) -> (Rel r -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (r o. r) = r))))
40	39	ex 402	. . . . . . . . . . . . 13 |- (Rel r -> (A.x e. U.U.rxrx -> (Rel r -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (r o. r) = r)))))
41	40	pm2.43a 80	. . . . . . . . . . . 12 |- (Rel r -> (A.x e. U.U.rxrx -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (r o. r) = r))))
42	41	com4l 43	. . . . . . . . . . 11 |- (A.x e. U.U.rxrx -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (Rel r -> (r o. r) = r))))
43	5, 42	sylbir 218	. . . . . . . . . 10 |- (( _I |` U.U.r) C_ r -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (Rel r -> (r o. r) = r))))
44	43	adantr 425	. . . . . . . . 9 |- ((( _I |` U.U.r) C_ r /\ ( _I |` U.U.r) C_ `'r) -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (Rel r -> (r o. r) = r))))
45	4, 44	sylbir 218	. . . . . . . 8 |- (( _I |` U.U.r) C_ (r i^i `'r) -> ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (Rel r -> (r o. r) = r))))
46	3, 45	mpcom 60	. . . . . . 7 |- ((r i^i `'r) = ( _I |` U.U.r) -> ((r o. r) C_ r -> (Rel r -> (r o. r) = r)))
47	46	com13 37	. . . . . 6 |- (Rel r -> ((r o. r) C_ r -> ((r i^i `'r) = ( _I |` U.U.r) -> (r o. r) = r)))
48	47	3imp 1061	. . . . 5 |- ((Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r)) -> (r o. r) = r)
49		simp3 878	. . . . 5 |- ((Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r)) -> (r i^i `'r) = ( _I |` U.U.r))
50	2, 48, 49	3jca 1050	. . . 4 |- ((Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r)) -> (Rel r /\ (r o. r) = r /\ (r i^i `'r) = ( _I |` U.U.r)))
51		simp1 876	. . . . 5 |- ((Rel r /\ (r o. r) = r /\ (r i^i `'r) = ( _I |` U.U.r)) -> Rel r)
52		eqimss 2665	. . . . . 6 |- ((r o. r) = r -> (r o. r) C_ r)
53	52	3ad2ant2 898	. . . . 5 |- ((Rel r /\ (r o. r) = r /\ (r i^i `'r) = ( _I |` U.U.r)) -> (r o. r) C_ r)
54		simp3 878	. . . . 5 |- ((Rel r /\ (r o. r) = r /\ (r i^i `'r) = ( _I |` U.U.r)) -> (r i^i `'r) = ( _I |` U.U.r))
55	51, 53, 54	3jca 1050	. . . 4 |- ((Rel r /\ (r o. r) = r /\ (r i^i `'r) = ( _I |` U.U.r)) -> (Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r)))
56	50, 55	impbii 174	. . 3 |- ((Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r)) <-> (Rel r /\ (r o. r) = r /\ (r i^i `'r) = ( _I |` U.U.r)))
57	56	abbii 2006	. 2 |- {r | (Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r))} = {r | (Rel r /\ (r o. r) = r /\ (r i^i `'r) = ( _I |` U.U.r))}
58	1, 57	eqtri 1908	1 |- Poset = {r | (Rel r /\ (r o. r) = r /\ (r i^i `'r) = ( _I |` U.U.r))}

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298  {cab 1871  A.wral 2105   i^i cin 2592   C_ wss 2593  U.cuni 3177   class class class wbr 3338   _I cid 3582  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988   o. ccom 3990  Rel wrel 3991  Posetcps 9980

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-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

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-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ps 9984

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