Theorem asymref2 3497

Description: Two ways of saying a relation is antisymmetric and reflexive.

Assertion

Ref	Expression
asymref2	|- ((R i^i `'R) = (I |` U.U.R) <-> (A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)))

Distinct variable group:   x,y,R

Proof of Theorem asymref2

Step	Hyp	Ref	Expression
1		df-ral 1696	. . 3 |- (A.x e. U.U.RA.y((xRy /\ yRx) <-> x = y) <-> A.x(x e. U.U.R -> A.y((xRy /\ yRx) <-> x = y)))
2		breq2 2678	. . . . . . . . . . . . 13 |- (y = x -> (xRy <-> xRx))
3		breq1 2677	. . . . . . . . . . . . 13 |- (y = x -> (yRx <-> xRx))
4	2, 3	anbi12d 639	. . . . . . . . . . . 12 |- (y = x -> ((xRy /\ yRx) <-> (xRx /\ xRx)))
5		anidm 442	. . . . . . . . . . . 12 |- ((xRx /\ xRx) <-> xRx)
6	4, 5	syl6bb 547	. . . . . . . . . . 11 |- (y = x -> ((xRy /\ yRx) <-> xRx))
7		equequ2 1177	. . . . . . . . . . 11 |- (y = x -> (x = y <-> x = x))
8	6, 7	bibi12d 640	. . . . . . . . . 10 |- (y = x -> (((xRy /\ yRx) <-> x = y) <-> (xRx <-> x = x)))
9		equid 1167	. . . . . . . . . . 11 |- x = x
10	9	tbt 732	. . . . . . . . . 10 |- (xRx <-> (xRx <-> x = x))
11	8, 10	syl6bbr 549	. . . . . . . . 9 |- (y = x -> (((xRy /\ yRx) <-> x = y) <-> xRx))
12	11	a4v 1314	. . . . . . . 8 |- (A.y((xRy /\ yRx) <-> x = y) -> xRx)
13		bi1 155	. . . . . . . . 9 |- (((xRy /\ yRx) <-> x = y) -> ((xRy /\ yRx) -> x = y))
14	13	19.20i 1033	. . . . . . . 8 |- (A.y((xRy /\ yRx) <-> x = y) -> A.y((xRy /\ yRx) -> x = y))
15	12, 14	jca 295	. . . . . . 7 |- (A.y((xRy /\ yRx) <-> x = y) -> (xRx /\ A.y((xRy /\ yRx) -> x = y)))
16		bi3 157	. . . . . . . . . 10 |- (((xRy /\ yRx) -> x = y) -> ((x = y -> (xRy /\ yRx)) -> ((xRy /\ yRx) <-> x = y)))
17		breq2 2678	. . . . . . . . . . . 12 |- (x = y -> (xRx <-> xRy))
18	17	biimpcd 162	. . . . . . . . . . 11 |- (xRx -> (x = y -> xRy))
19		breq1 2677	. . . . . . . . . . . 12 |- (x = y -> (xRx <-> yRx))
20	19	biimpcd 162	. . . . . . . . . . 11 |- (xRx -> (x = y -> yRx))
21	18, 20	jcad 611	. . . . . . . . . 10 |- (xRx -> (x = y -> (xRy /\ yRx)))
22	16, 21	syl5com 52	. . . . . . . . 9 |- (xRx -> (((xRy /\ yRx) -> x = y) -> ((xRy /\ yRx) <-> x = y)))
23	22	19.20dv 1331	. . . . . . . 8 |- (xRx -> (A.y((xRy /\ yRx) -> x = y) -> A.y((xRy /\ yRx) <-> x = y)))
24	23	imp 357	. . . . . . 7 |- ((xRx /\ A.y((xRy /\ yRx) -> x = y)) -> A.y((xRy /\ yRx) <-> x = y))
25	15, 24	impbii 164	. . . . . 6 |- (A.y((xRy /\ yRx) <-> x = y) <-> (xRx /\ A.y((xRy /\ yRx) -> x = y)))
26	25	imbi2i 192	. . . . 5 |- ((x e. U.U.R -> A.y((xRy /\ yRx) <-> x = y)) <-> (x e. U.U.R -> (xRx /\ A.y((xRy /\ yRx) -> x = y))))
27		pm4.76 610	. . . . 5 |- (((x e. U.U.R -> xRx) /\ (x e. U.U.R -> A.y((xRy /\ yRx) -> x = y))) <-> (x e. U.U.R -> (xRx /\ A.y((xRy /\ yRx) -> x = y))))
28		visset 1860	. . . . . . . . . . . . . 14 |- x e. V
29	28	breldm 3372	. . . . . . . . . . . . 13 |- (xRy -> x e. dom R)
30		ssun1 2244	. . . . . . . . . . . . . . 15 |- dom R (_ (dom R u. ran R)
31		dmrnssfld 3414	. . . . . . . . . . . . . . 15 |- (dom R u. ran R) (_ U.U.R
32	30, 31	sstri 2124	. . . . . . . . . . . . . 14 |- dom R (_ U.U.R
33	32	sseli 2116	. . . . . . . . . . . . 13 |- (x e. dom R -> x e. U.U.R)
34	29, 33	syl 10	. . . . . . . . . . . 12 |- (xRy -> x e. U.U.R)
35	34	adantr 398	. . . . . . . . . . 11 |- ((xRy /\ yRx) -> x e. U.U.R)
36	35	pm4.71ri 649	. . . . . . . . . 10 |- ((xRy /\ yRx) <-> (x e. U.U.R /\ (xRy /\ yRx)))
37	36	imbi1i 193	. . . . . . . . 9 |- (((xRy /\ yRx) -> x = y) <-> ((x e. U.U.R /\ (xRy /\ yRx)) -> x = y))
38		impexp 354	. . . . . . . . 9 |- (((x e. U.U.R /\ (xRy /\ yRx)) -> x = y) <-> (x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
39	37, 38	bitri 180	. . . . . . . 8 |- (((xRy /\ yRx) -> x = y) <-> (x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
40	39	albii 1040	. . . . . . 7 |- (A.y((xRy /\ yRx) -> x = y) <-> A.y(x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
41		19.21v 1327	. . . . . . 7 |- (A.y(x e. U.U.R -> ((xRy /\ yRx) -> x = y)) <-> (x e. U.U.R -> A.y((xRy /\ yRx) -> x = y)))
42	40, 41	bitr2i 181	. . . . . 6 |- ((x e. U.U.R -> A.y((xRy /\ yRx) -> x = y)) <-> A.y((xRy /\ yRx) -> x = y))
43	42	anbi2i 491	. . . . 5 |- (((x e. U.U.R -> xRx) /\ (x e. U.U.R -> A.y((xRy /\ yRx) -> x = y))) <-> ((x e. U.U.R -> xRx) /\ A.y((xRy /\ yRx) -> x = y)))
44	26, 27, 43	3bitr2i 186	. . . 4 |- ((x e. U.U.R -> A.y((xRy /\ yRx) <-> x = y)) <-> ((x e. U.U.R -> xRx) /\ A.y((xRy /\ yRx) -> x = y)))
45	44	albii 1040	. . 3 |- (A.x(x e. U.U.R -> A.y((xRy /\ yRx) <-> x = y)) <-> A.x((x e. U.U.R -> xRx) /\ A.y((xRy /\ yRx) -> x = y)))
46		19.26 1108	. . 3 |- (A.x((x e. U.U.R -> xRx) /\ A.y((xRy /\ yRx) -> x = y)) <-> (A.x(x e. U.U.R -> xRx) /\ A.xA.y((xRy /\ yRx) -> x = y)))
47	1, 45, 46	3bitri 184	. 2 |- (A.x e. U.U.RA.y((xRy /\ yRx) <-> x = y) <-> (A.x(x e. U.U.R -> xRx) /\ A.xA.y((xRy /\ yRx) -> x = y)))
48		asymref 3496	. 2 |- ((R i^i `'R) = (I |` U.U.R) <-> A.x e. U.U.RA.y((xRy /\ yRx) <-> x = y))
49		df-ral 1696	. . 3 |- (A.x e. U.U.RxRx <-> A.x(x e. U.U.R -> xRx))
50	49	anbi1i 492	. 2 |- ((A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)) <-> (A.x(x e. U.U.R -> xRx) /\ A.xA.y((xRy /\ yRx) -> x = y)))
51	47, 48, 50	3bitr4i 190	1 |- ((R i^i `'R) = (I |` U.U.R) <-> (A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230  A.wal 995   = wceq 997   e. wcel 999  A.wral 1692   u. cun 2096   i^i cin 2097  U.cuni 2557   class class class wbr 2674  Icid 2887  `'ccnv 3226  dom cdm 3227  ran crn 3228   |` cres 3229

This theorem is referenced by:  pslem 8731

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-id 2891  df-xp 3241  df-rel 3242  df-cnv 3243  df-dm 3245  df-rn 3246  df-res 3247

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