Theorem asymref2 3532

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 1695	. . 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 2696	. . . . . . . . . . . . 13 |- (y = x -> (xRy <-> xRx))
3		breq1 2695	. . . . . . . . . . . . 13 |- (y = x -> (yRx <-> xRx))
4	2, 3	anbi12d 631	. . . . . . . . . . . 12 |- (y = x -> ((xRy /\ yRx) <-> (xRx /\ xRx)))
5		anidm 433	. . . . . . . . . . . 12 |- ((xRx /\ xRx) <-> xRx)
6	4, 5	syl6bb 539	. . . . . . . . . . 11 |- (y = x -> ((xRy /\ yRx) <-> xRx))
7		equequ2 1172	. . . . . . . . . . 11 |- (y = x -> (x = y <-> x = x))
8	6, 7	bibi12d 632	. . . . . . . . . 10 |- (y = x -> (((xRy /\ yRx) <-> x = y) <-> (xRx <-> x = x)))
9		equid 1162	. . . . . . . . . . 11 |- x = x
10	9	tbt 725	. . . . . . . . . 10 |- (xRx <-> (xRx <-> x = x))
11	8, 10	syl6bbr 541	. . . . . . . . 9 |- (y = x -> (((xRy /\ yRx) <-> x = y) <-> xRx))
12	11	a4v 1310	. . . . . . . 8 |- (A.y((xRy /\ yRx) <-> x = y) -> xRx)
13		bi1 146	. . . . . . . . 9 |- (((xRy /\ yRx) <-> x = y) -> ((xRy /\ yRx) -> x = y))
14	13	19.20i 1028	. . . . . . . 8 |- (A.y((xRy /\ yRx) <-> x = y) -> A.y((xRy /\ yRx) -> x = y))
15	12, 14	jca 286	. . . . . . 7 |- (A.y((xRy /\ yRx) <-> x = y) -> (xRx /\ A.y((xRy /\ yRx) -> x = y)))
16		bi3 148	. . . . . . . . . 10 |- (((xRy /\ yRx) -> x = y) -> ((x = y -> (xRy /\ yRx)) -> ((xRy /\ yRx) <-> x = y)))
17		breq2 2696	. . . . . . . . . . . 12 |- (x = y -> (xRx <-> xRy))
18	17	biimpcd 153	. . . . . . . . . . 11 |- (xRx -> (x = y -> xRy))
19		breq1 2695	. . . . . . . . . . . 12 |- (x = y -> (xRx <-> yRx))
20	19	biimpcd 153	. . . . . . . . . . 11 |- (xRx -> (x = y -> yRx))
21	18, 20	jcad 603	. . . . . . . . . 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 1327	. . . . . . . 8 |- (xRx -> (A.y((xRy /\ yRx) -> x = y) -> A.y((xRy /\ yRx) <-> x = y)))
24	23	imp 348	. . . . . . 7 |- ((xRx /\ A.y((xRy /\ yRx) -> x = y)) -> A.y((xRy /\ yRx) <-> x = y))
25	15, 24	impbii 155	. . . . . 6 |- (A.y((xRy /\ yRx) <-> x = y) <-> (xRx /\ A.y((xRy /\ yRx) -> x = y)))
26	25	imbi2i 183	. . . . 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		pm4.76 602	. . . . 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 1859	. . . . . . . . . . . . . 14 |- x e. V
29	28	breldm 3406	. . . . . . . . . . . . 13 |- (xRy -> x e. dom R)
30		ssun1 2245	. . . . . . . . . . . . . . 15 |- dom R (_ (dom R u. ran R)
31		dmrnssfld 3444	. . . . . . . . . . . . . . 15 |- (dom R u. ran R) (_ U.U.R
32	30, 31	sstri 2125	. . . . . . . . . . . . . 14 |- dom R (_ U.U.R
33	32	sseli 2117	. . . . . . . . . . . . 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 389	. . . . . . . . . . 11 |- ((xRy /\ yRx) -> x e. U.U.R)
36	35	pm4.71ri 641	. . . . . . . . . 10 |- ((xRy /\ yRx) <-> (x e. U.U.R /\ (xRy /\ yRx)))
37	36	imbi1i 184	. . . . . . . . 9 |- (((xRy /\ yRx) -> x = y) <-> ((x e. U.U.R /\ (xRy /\ yRx)) -> x = y))
38		impexp 345	. . . . . . . . 9 |- (((x e. U.U.R /\ (xRy /\ yRx)) -> x = y) <-> (x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
39	37, 38	bitri 171	. . . . . . . 8 |- (((xRy /\ yRx) -> x = y) <-> (x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
40	39	albii 1035	. . . . . . 7 |- (A.y((xRy /\ yRx) -> x = y) <-> A.y(x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
41		19.21v 1323	. . . . . . 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 172	. . . . . 6 |- ((x e. U.U.R -> A.y((xRy /\ yRx) -> x = y)) <-> A.y((xRy /\ yRx) -> x = y))
43	42	anbi2i 483	. . . . 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 177	. . . 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 1035	. . 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 1103	. . 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 175	. 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 3531	. 2 |- ((R i^i `'R) = (I |` U.U.R) <-> A.x e. U.U.RA.y((xRy /\ yRx) <-> x = y))
49		df-ral 1695	. . 3 |- (A.x e. U.U.RxRx <-> A.x(x e. U.U.R -> xRx))
50	49	anbi1i 484	. 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 181	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 144   /\ wa 221  A.wal 990   = wceq 992   e. wcel 994  A.wral 1691   u. cun 2097   i^i cin 2098  U.cuni 2569   class class class wbr 2692  Icid 2909  `'ccnv 3250  dom cdm 3251  ran crn 3252   |` cres 3253

This theorem is referenced by:  pslem 8909  pospos 10882

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-dm 3269  df-rn 3270  df-res 3271

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