Theorem asymref2 3440

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 1649	. . 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 2623	. . . . . . . . . . . . 13 |- (y = x -> (xRy <-> xRx))
3		breq1 2622	. . . . . . . . . . . . 13 |- (y = x -> (yRx <-> xRx))
4	2, 3	anbi12d 628	. . . . . . . . . . . 12 |- (y = x -> ((xRy /\ yRx) <-> (xRx /\ xRx)))
5		anidm 432	. . . . . . . . . . . 12 |- ((xRx /\ xRx) <-> xRx)
6	4, 5	syl6bb 536	. . . . . . . . . . 11 |- (y = x -> ((xRy /\ yRx) <-> xRx))
7		equequ2 1135	. . . . . . . . . . 11 |- (y = x -> (x = y <-> x = x))
8	6, 7	bibi12d 629	. . . . . . . . . 10 |- (y = x -> (((xRy /\ yRx) <-> x = y) <-> (xRx <-> x = x)))
9		equid 1126	. . . . . . . . . . 11 |- x = x
10	9	tbt 720	. . . . . . . . . 10 |- (xRx <-> (xRx <-> x = x))
11	8, 10	syl6bbr 538	. . . . . . . . 9 |- (y = x -> (((xRy /\ yRx) <-> x = y) <-> xRx))
12	11	a4v 1272	. . . . . . . 8 |- (A.y((xRy /\ yRx) <-> x = y) -> xRx)
13		bi1 148	. . . . . . . . 9 |- (((xRy /\ yRx) <-> x = y) -> ((xRy /\ yRx) -> x = y))
14	13	19.20i 992	. . . . . . . 8 |- (A.y((xRy /\ yRx) <-> x = y) -> A.y((xRy /\ yRx) -> x = y))
15	12, 14	jca 288	. . . . . . 7 |- (A.y((xRy /\ yRx) <-> x = y) -> (xRx /\ A.y((xRy /\ yRx) -> x = y)))
16		bi3 150	. . . . . . . . . 10 |- (((xRy /\ yRx) -> x = y) -> ((x = y -> (xRy /\ yRx)) -> ((xRy /\ yRx) <-> x = y)))
17		breq2 2623	. . . . . . . . . . . 12 |- (x = y -> (xRx <-> xRy))
18	17	biimpcd 155	. . . . . . . . . . 11 |- (xRx -> (x = y -> xRy))
19		breq1 2622	. . . . . . . . . . . 12 |- (x = y -> (xRx <-> yRx))
20	19	biimpcd 155	. . . . . . . . . . 11 |- (xRx -> (x = y -> yRx))
21	18, 20	jcad 600	. . . . . . . . . 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 1289	. . . . . . . 8 |- (xRx -> (A.y((xRy /\ yRx) -> x = y) -> A.y((xRy /\ yRx) <-> x = y)))
24	23	imp 350	. . . . . . 7 |- ((xRx /\ A.y((xRy /\ yRx) -> x = y)) -> A.y((xRy /\ yRx) <-> x = y))
25	15, 24	impbi 157	. . . . . 6 |- (A.y((xRy /\ yRx) <-> x = y) <-> (xRx /\ A.y((xRy /\ yRx) -> x = y)))
26	25	imbi2i 185	. . . . 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		pm4.76 599	. . . . 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 1813	. . . . . . . . . . . . . 14 |- x e. V
29	28	breldm 3315	. . . . . . . . . . . . 13 |- (xRy -> x e. dom R)
30		ssun1 2193	. . . . . . . . . . . . . . 15 |- dom R (_ (dom R u. ran R)
31		dmrnssfld 3357	. . . . . . . . . . . . . . 15 |- (dom R u. ran R) (_ U.U.R
32	30, 31	sstri 2073	. . . . . . . . . . . . . 14 |- dom R (_ U.U.R
33	32	sseli 2065	. . . . . . . . . . . . 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 638	. . . . . . . . . 10 |- ((xRy /\ yRx) <-> (x e. U.U.R /\ (xRy /\ yRx)))
37	36	imbi1i 186	. . . . . . . . 9 |- (((xRy /\ yRx) -> x = y) <-> ((x e. U.U.R /\ (xRy /\ yRx)) -> x = y))
38		impexp 347	. . . . . . . . 9 |- (((x e. U.U.R /\ (xRy /\ yRx)) -> x = y) <-> (x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
39	37, 38	bitr 173	. . . . . . . 8 |- (((xRy /\ yRx) -> x = y) <-> (x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
40	39	albii 999	. . . . . . 7 |- (A.y((xRy /\ yRx) -> x = y) <-> A.y(x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
41		19.21v 1285	. . . . . . 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	bitr2 174	. . . . . 6 |- ((x e. U.U.R -> A.y((xRy /\ yRx) -> x = y)) <-> A.y((xRy /\ yRx) -> x = y))
43	42	anbi2i 480	. . . . 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	3bitr2 179	. . . 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 999	. . 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 1067	. . 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	3bitr 177	. 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 3439	. 2 |- ((R i^i `'R) = (I |` U.U.R) <-> A.x e. U.U.RA.y((xRy /\ yRx) <-> x = y))
49		df-ral 1649	. . 3 |- (A.x e. U.U.RxRx <-> A.x(x e. U.U.R -> xRx))
50	49	anbi1i 481	. 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	3bitr4 183	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 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  A.wral 1645   u. cun 2045   i^i cin 2046  U.cuni 2503   class class class wbr 2619  Icid 2831  `'ccnv 3169  dom cdm 3170  ran crn 3171   |` cres 3172

This theorem is referenced by:  pslem 8647

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-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-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190

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