Theorem erreft 16259

Description: Closed form of erref 5333.

Assertion

Ref	Expression
erreft	|- (Er R -> (A e. (dom R u. ran R) -> ARA))

Proof of Theorem erreft

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . . . . . 10 |- x e. _V
2		visset 2295	. . . . . . . . . 10 |- y e. _V
3	1, 2, 1	ertr2 16257	. . . . . . . . 9 |- (Er R -> ((xRy /\ yRx) -> xRx))
4	1, 2	ersym2 16256	. . . . . . . . 9 |- (Er R -> (xRy -> yRx))
5	3, 4	sylan2d 507	. . . . . . . 8 |- (Er R -> ((xRy /\ xRy) -> xRx))
6		anidm 478	. . . . . . . 8 |- ((xRy /\ xRy) <-> xRy)
7	5, 6	syl5ibr 224	. . . . . . 7 |- (Er R -> (xRy -> xRx))
8	7	19.23adv 1584	. . . . . 6 |- (Er R -> (E.y xRy -> xRx))
9	1	eldm 4153	. . . . . 6 |- (x e. dom R <-> E.y xRy)
10	8, 9	syl5ib 223	. . . . 5 |- (Er R -> (x e. dom R -> xRx))
11	2, 1	ersym2 16256	. . . . . . . . 9 |- (Er R -> (yRx -> xRy))
12	3, 11	syland 506	. . . . . . . 8 |- (Er R -> ((yRx /\ yRx) -> xRx))
13		anidm 478	. . . . . . . 8 |- ((yRx /\ yRx) <-> yRx)
14	12, 13	syl5ibr 224	. . . . . . 7 |- (Er R -> (yRx -> xRx))
15	14	19.23adv 1584	. . . . . 6 |- (Er R -> (E.y yRx -> xRx))
16	1	elrn 4197	. . . . . 6 |- (x e. ran R <-> E.y yRx)
17	15, 16	syl5ib 223	. . . . 5 |- (Er R -> (x e. ran R -> xRx))
18	10, 17	jaod 469	. . . 4 |- (Er R -> ((x e. dom R \/ x e. ran R) -> xRx))
19		elun 2741	. . . 4 |- (x e. (dom R u. ran R) <-> (x e. dom R \/ x e. ran R))
20	18, 19	syl5ib 223	. . 3 |- (Er R -> (x e. (dom R u. ran R) -> xRx))
21	20	r19.21aiv 2175	. 2 |- (Er R -> A.x e. (dom R u. ran R)xRx)
22		breq1 3341	. . . 4 |- (x = A -> (xRx <-> ARx))
23		breq2 3342	. . . 4 |- (x = A -> (ARx <-> ARA))
24	22, 23	bitrd 587	. . 3 |- (x = A -> (xRx <-> ARA))
25	24	rcla4cv 2377	. 2 |- (A.x e. (dom R u. ran R)xRx -> (A e. (dom R u. ran R) -> ARA))
26	21, 25	syl 12	1 |- (Er R -> (A e. (dom R u. ran R) -> ARA))

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105   u. cun 2591   class class class wbr 3338  dom cdm 3986  ran crn 3987  Er wer 5315

This theorem is referenced by:  erreft2 16261

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-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-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-er 5318

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