Theorem asymref2OLD 4311

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

Assertion

Ref	Expression
asymref2OLD	|- ((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 asymref2OLD

Step	Hyp	Ref	Expression
1		df-ral 2109	. . 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 3342	. . . . . . . . . . . . 13 |- (y = x -> (xRy <-> xRx))
3		breq1 3341	. . . . . . . . . . . . 13 |- (y = x -> (yRx <-> xRx))
4	2, 3	anbi12d 690	. . . . . . . . . . . 12 |- (y = x -> ((xRy /\ yRx) <-> (xRx /\ xRx)))
5		anidm 478	. . . . . . . . . . . 12 |- ((xRx /\ xRx) <-> xRx)
6	4, 5	syl6bb 595	. . . . . . . . . . 11 |- (y = x -> ((xRy /\ yRx) <-> xRx))
7		equequ2 1495	. . . . . . . . . . 11 |- (y = x -> (x = y <-> x = x))
8	6, 7	bibi12d 691	. . . . . . . . . 10 |- (y = x -> (((xRy /\ yRx) <-> x = y) <-> (xRx <-> x = x)))
9		equid 1484	. . . . . . . . . . 11 |- x = x
10	9	tbt 788	. . . . . . . . . 10 |- (xRx <-> (xRx <-> x = x))
11	8, 10	syl6bbr 597	. . . . . . . . 9 |- (y = x -> (((xRy /\ yRx) <-> x = y) <-> xRx))
12	11	a4v 1649	. . . . . . . 8 |- (A.y((xRy /\ yRx) <-> x = y) -> xRx)
13		bi1 165	. . . . . . . . 9 |- (((xRy /\ yRx) <-> x = y) -> ((xRy /\ yRx) -> x = y))
14	13	alimi 1338	. . . . . . . 8 |- (A.y((xRy /\ yRx) <-> x = y) -> A.y((xRy /\ yRx) -> x = y))
15	12, 14	jca 310	. . . . . . 7 |- (A.y((xRy /\ yRx) <-> x = y) -> (xRx /\ A.y((xRy /\ yRx) -> x = y)))
16		bi3 167	. . . . . . . . . 10 |- (((xRy /\ yRx) -> x = y) -> ((x = y -> (xRy /\ yRx)) -> ((xRy /\ yRx) <-> x = y)))
17		breq2 3342	. . . . . . . . . . . 12 |- (x = y -> (xRx <-> xRy))
18	17	biimpcd 172	. . . . . . . . . . 11 |- (xRx -> (x = y -> xRy))
19		breq1 3341	. . . . . . . . . . . 12 |- (x = y -> (xRx <-> yRx))
20	19	biimpcd 172	. . . . . . . . . . 11 |- (xRx -> (x = y -> yRx))
21	18, 20	jcad 661	. . . . . . . . . 10 |- (xRx -> (x = y -> (xRy /\ yRx)))
22	16, 21	syl5com 63	. . . . . . . . 9 |- (xRx -> (((xRy /\ yRx) -> x = y) -> ((xRy /\ yRx) <-> x = y)))
23	22	alimdv 1668	. . . . . . . 8 |- (xRx -> (A.y((xRy /\ yRx) -> x = y) -> A.y((xRy /\ yRx) <-> x = y)))
24	23	imp 377	. . . . . . 7 |- ((xRx /\ A.y((xRy /\ yRx) -> x = y)) -> A.y((xRy /\ yRx) <-> x = y))
25	15, 24	impbii 174	. . . . . 6 |- (A.y((xRy /\ yRx) <-> x = y) <-> (xRx /\ A.y((xRy /\ yRx) -> x = y)))
26	25	imbi2i 202	. . . . 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		pm4.76 660	. . . . 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 2295	. . . . . . . . . . . . . 14 |- x e. _V
29	28	breldm 4161	. . . . . . . . . . . . 13 |- (xRy -> x e. dom R)
30		ssun1 2767	. . . . . . . . . . . . . . 15 |- dom R C_ (dom R u. ran R)
31		dmrnssfld 4205	. . . . . . . . . . . . . . 15 |- (dom R u. ran R) C_ U.U.R
32	30, 31	sstri 2626	. . . . . . . . . . . . . 14 |- dom R C_ U.U.R
33	32	sseli 2617	. . . . . . . . . . . . 13 |- (x e. dom R -> x e. U.U.R)
34	29, 33	syl 12	. . . . . . . . . . . 12 |- (xRy -> x e. U.U.R)
35	34	adantr 425	. . . . . . . . . . 11 |- ((xRy /\ yRx) -> x e. U.U.R)
36	35	pm4.71ri 700	. . . . . . . . . 10 |- ((xRy /\ yRx) <-> (x e. U.U.R /\ (xRy /\ yRx)))
37	36	imbi1i 203	. . . . . . . . 9 |- (((xRy /\ yRx) -> x = y) <-> ((x e. U.U.R /\ (xRy /\ yRx)) -> x = y))
38		impexp 374	. . . . . . . . 9 |- (((x e. U.U.R /\ (xRy /\ yRx)) -> x = y) <-> (x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
39	37, 38	bitri 190	. . . . . . . 8 |- (((xRy /\ yRx) -> x = y) <-> (x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
40	39	albii 1346	. . . . . . 7 |- (A.y((xRy /\ yRx) -> x = y) <-> A.y(x e. U.U.R -> ((xRy /\ yRx) -> x = y)))
41		19.21v 1663	. . . . . . 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 191	. . . . . 6 |- ((x e. U.U.R -> A.y((xRy /\ yRx) -> x = y)) <-> A.y((xRy /\ yRx) -> x = y))
43	42	anbi2i 538	. . . . 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 196	. . . 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 1346	. . 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 1416	. . 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 194	. 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 4308	. 2 |- ((R i^i `'R) = ( _I |` U.U.R) <-> A.x e. U.U.RA.y((xRy /\ yRx) <-> x = y))
49		df-ral 2109	. . 3 |- (A.x e. U.U.RxRx <-> A.x(x e. U.U.R -> xRx))
50	49	anbi1i 539	. 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 200	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 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105   u. cun 2591   i^i cin 2592  U.cuni 3177   class class class wbr 3338   _I cid 3582  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988

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-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-dm 4004  df-rn 4005  df-res 4006

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