Theorem erref 4272

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 2619	. . 3 |- (x = A -> (xRx <-> ARx))
2		breq2 2620	. . 3 |- (x = A -> (ARx <-> ARA))
3	1, 2	bitrd 527	. 2 |- (x = A -> (xRx <-> ARA))
4		elun 2171	. . 3 |- (x e. (dom R u. ran R) <-> (x e. dom R \/ x e. ran R))
5		visset 1811	. . . . . 6 |- x e. V
6	5	eldm 3304	. . . . 5 |- (x e. dom R <-> E.y xRy)
7		visset 1811	. . . . . . . . 9 |- y e. V
8		erref.1	. . . . . . . . 9 |- Er R
9	5, 7, 5, 8	ertr 4271	. . . . . . . 8 |- ((xRy /\ yRx) -> xRx)
10	5, 7, 8	ersymb 4270	. . . . . . . 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 3347	. . . . 5 |- (x e. ran R <-> E.y yRx)
16	7, 5, 8	ersymb 4270	. . . . . . . 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 1850	1 |- (A e. (dom R u. ran R) -> ARA)

Syntax hints:   -> wi 3   \/ wo 222   = wceq 955   e. wcel 957  E.wex 979   u. cun 2043   class class class wbr 2616  dom cdm 3167  ran crn 3168  Er wer 4255

This theorem is referenced by:  erth 4279

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-er 4258

Copyright terms: Public domain