Theorem erref 3212

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 2065	. . 3 |- (x = A -> (xRx <-> ARx))
2		breq2 2066	. . 3 |- (x = A -> (ARx <-> ARA))
3	1, 2	bitrd 406	. 2 |- (x = A -> (xRx <-> ARA))
4		elun 1601	. . 3 |- (x e. (dom R u. ran R) <-> (x e. dom R \/ x e. ran R))
5		visset 1350	. . . . . 6 |- x e. V
6	5	eldm 2527	. . . . 5 |- (x e. dom R <-> E.y xRy)
7		visset 1350	. . . . . . . . 9 |- y e. V
8		erref.1	. . . . . . . . 9 |- Er R
9	5, 7, 5, 8	ertr 3211	. . . . . . . 8 |- ((xRy /\ yRx) -> xRx)
10	5, 7, 8	ersymb 3210	. . . . . . . 8 |- (xRy <-> yRx)
11	9, 10	sylan2b 347	. . . . . . 7 |- ((xRy /\ xRy) -> xRx)
12	11	anidms 332	. . . . . 6 |- (xRy -> xRx)
13	12	19.23aiv 952	. . . . 5 |- (E.y xRy -> xRx)
14	6, 13	sylbi 174	. . . 4 |- (x e. dom R -> xRx)
15	5	elrn2 2563	. . . . 5 |- (x e. ran R <-> E.y yRx)
16	7, 5, 8	ersymb 3210	. . . . . . . 8 |- (yRx <-> xRy)
17	9, 16	sylanb 344	. . . . . . 7 |- ((yRx /\ yRx) -> xRx)
18	17	anidms 332	. . . . . 6 |- (yRx -> xRx)
19	18	19.23aiv 952	. . . . 5 |- (E.y yRx -> xRx)
20	15, 19	sylbi 174	. . . 4 |- (x e. ran R -> xRx)
21	14, 20	jaoi 275	. . 3 |- ((x e. dom R \/ x e. ran R) -> xRx)
22	4, 21	sylbi 174	. 2 |- (x e. (dom R u. ran R) -> xRx)
23	3, 22	vtoclga 1387	1 |- (A e. (dom R u. ran R) -> ARA)

Syntax hints:   -> wi 2   \/ wo 195  E.wex 678   = wceq 1091   e. wcel 1092   u. cun 1485   class class class wbr 2054  dom cdm 2410  ran crn 2411  Er wer 3197

This theorem is referenced by:  erth 3219

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-er 3200

metamath.org