Theorem erref 4275

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 2622	. . 3 |- (x = A -> (xRx <-> ARx))
2		breq2 2623	. . 3 |- (x = A -> (ARx <-> ARA))
3	1, 2	bitrd 528	. 2 |- (x = A -> (xRx <-> ARA))
4		elun 2173	. . 3 |- (x e. (dom R u. ran R) <-> (x e. dom R \/ x e. ran R))
5		visset 1813	. . . . . 6 |- x e. V
6	5	eldm 3307	. . . . 5 |- (x e. dom R <-> E.y xRy)
7		visset 1813	. . . . . . . . 9 |- y e. V
8		erref.1	. . . . . . . . 9 |- Er R
9	5, 7, 5, 8	ertr 4274	. . . . . . . 8 |- ((xRy /\ yRx) -> xRx)
10	5, 7, 8	ersymb 4273	. . . . . . . 8 |- (xRy <-> yRx)
11	9, 10	sylan2b 452	. . . . . . 7 |- ((xRy /\ xRy) -> xRx)
12	11	anidms 434	. . . . . 6 |- (xRy -> xRx)
13	12	19.23aiv 1295	. . . . 5 |- (E.y xRy -> xRx)
14	6, 13	sylbi 199	. . . 4 |- (x e. dom R -> xRx)
15	5	elrn 3350	. . . . 5 |- (x e. ran R <-> E.y yRx)
16	7, 5, 8	ersymb 4273	. . . . . . . 8 |- (yRx <-> xRy)
17	9, 16	sylanb 449	. . . . . . 7 |- ((yRx /\ yRx) -> xRx)
18	17	anidms 434	. . . . . 6 |- (yRx -> xRx)
19	18	19.23aiv 1295	. . . . 5 |- (E.y yRx -> xRx)
20	15, 19	sylbi 199	. . . 4 |- (x e. ran R -> xRx)
21	14, 20	jaoi 341	. . 3 |- ((x e. dom R \/ x e. ran R) -> xRx)
22	4, 21	sylbi 199	. 2 |- (x e. (dom R u. ran R) -> xRx)
23	3, 22	vtoclga 1852	1 |- (A e. (dom R u. ran R) -> ARA)

Syntax hints:   -> wi 3   \/ wo 222   = wceq 956   e. wcel 958  E.wex 980   u. cun 2045   class class class wbr 2619  dom cdm 3170  ran crn 3171  Er wer 4258

This theorem is referenced by:  erth 4282

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-9 965  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-3an 777  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-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-br 2620  df-opab 2667  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-er 4261

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