Theorem erref 4265

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 2617	. . 3 |- (x = A -> (xRx <-> ARx))
2		breq2 2618	. . 3 |- (x = A -> (ARx <-> ARA))
3	1, 2	bitrd 527	. 2 |- (x = A -> (xRx <-> ARA))
4		elun 2169	. . 3 |- (x e. (dom R u. ran R) <-> (x e. dom R \/ x e. ran R))
5		visset 1809	. . . . . 6 |- x e. V
6	5	eldm 3302	. . . . 5 |- (x e. dom R <-> E.y xRy)
7		visset 1809	. . . . . . . . 9 |- y e. V
8		erref.1	. . . . . . . . 9 |- Er R
9	5, 7, 5, 8	ertr 4264	. . . . . . . 8 |- ((xRy /\ yRx) -> xRx)
10	5, 7, 8	ersymb 4263	. . . . . . . 8 |- (xRy <-> yRx)
11	9, 10	sylan2b 452	. . . . . . 7 |- ((xRy /\ xRy) -> xRx)
12	11	anidms 434	. . . . . 6 |- (xRy -> xRx)
13	12	19.23aiv 1293	. . . . 5 |- (E.y xRy -> xRx)
14	6, 13	sylbi 199	. . . 4 |- (x e. dom R -> xRx)
15	5	elrn 3344	. . . . 5 |- (x e. ran R <-> E.y yRx)
16	7, 5, 8	ersymb 4263	. . . . . . . 8 |- (yRx <-> xRy)
17	9, 16	sylanb 449	. . . . . . 7 |- ((yRx /\ yRx) -> xRx)
18	17	anidms 434	. . . . . 6 |- (yRx -> xRx)
19	18	19.23aiv 1293	. . . . 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 1848	1 |- (A e. (dom R u. ran R) -> ARA)

Syntax hints:   -> wi 3   \/ wo 222   = wceq 954   e. wcel 956  E.wex 978   u. cun 2041   class class class wbr 2614  dom cdm 3165  ran crn 3166  Er wer 4248

This theorem is referenced by:  erth 4272

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-er 4251

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