Theorem erref 4359

Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56.

Hypothesis

Ref	Expression
erref.1	|- Er R

Assertion

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

Proof of Theorem erref

Step	Hyp	Ref	Expression
1		breq1 2672	. . 3 |- (x = A -> (xRx <-> ARx))
2		breq2 2673	. . 3 |- (x = A -> (ARx <-> ARA))
3	1, 2	bitrd 530	. 2 |- (x = A -> (xRx <-> ARA))
4		elun 2217	. . 3 |- (x e. (dom R u. ran R) <-> (x e. dom R \/ x e. ran R))
5		visset 1851	. . . . . 6 |- x e. V
6	5	eldm 3371	. . . . 5 |- (x e. dom R <-> E.y xRy)
7		visset 1851	. . . . . . . . 9 |- y e. V
8		erref.1	. . . . . . . . 9 |- Er R
9	5, 7, 5, 8	ertr 4358	. . . . . . . 8 |- ((xRy /\ yRx) -> xRx)
10	5, 7, 8	ersymb 4357	. . . . . . . 8 |- (xRy <-> yRx)
11	9, 10	sylan2b 454	. . . . . . 7 |- ((xRy /\ xRy) -> xRx)
12	11	anidms 435	. . . . . 6 |- (xRy -> xRx)
13	12	19.23aiv 1328	. . . . 5 |- (E.y xRy -> xRx)
14	6, 13	sylbi 197	. . . 4 |- (x e. dom R -> xRx)
15	5	elrn 3410	. . . . 5 |- (x e. ran R <-> E.y yRx)
16	7, 5, 8	ersymb 4357	. . . . . . . 8 |- (yRx <-> xRy)
17	9, 16	sylanb 451	. . . . . . 7 |- ((yRx /\ yRx) -> xRx)
18	17	anidms 435	. . . . . 6 |- (yRx -> xRx)
19	18	19.23aiv 1328	. . . . 5 |- (E.y yRx -> xRx)
20	15, 19	sylbi 197	. . . 4 |- (x e. ran R -> xRx)
21	14, 20	jaoi 339	. . 3 |- ((x e. dom R \/ x e. ran R) -> xRx)
22	4, 21	sylbi 197	. 2 |- (x e. (dom R u. ran R) -> xRx)
23	3, 22	vtoclga 1890	1 |- (A e. (dom R u. ran R) -> ARA)

Syntax hints:   -> wi 3   \/ wo 220   = wceq 988   e. wcel 990  E.wex 1012   u. cun 2089   class class class wbr 2669  dom cdm 3225  ran crn 3226  Er wer 4342

This theorem is referenced by:  erth 4367

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-opab 2718  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-er 4345

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