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 ∈ (dom R ∪ 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 ∈ (dom R ∪ ran R) ↔ (x ∈ dom R ∨ x ∈ ran R))
5		visset 1350	. . . . . 6 ⊢ x ∈ V
6	5	eldm 2527	. . . . 5 ⊢ (x ∈ dom R ↔ ∃y xRy)
7		visset 1350	. . . . . . . . 9 ⊢ y ∈ 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 ⊢ (∃y xRy → xRx)
14	6, 13	sylbi 174	. . . 4 ⊢ (x ∈ dom R → xRx)
15	5	elrn2 2563	. . . . 5 ⊢ (x ∈ ran R ↔ ∃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 ⊢ (∃y yRx → xRx)
20	15, 19	sylbi 174	. . . 4 ⊢ (x ∈ ran R → xRx)
21	14, 20	jaoi 275	. . 3 ⊢ ((x ∈ dom R ∨ x ∈ ran R) → xRx)
22	4, 21	sylbi 174	. 2 ⊢ (x ∈ (dom R ∪ ran R) → xRx)
23	3, 22	vtoclga 1387	1 ⊢ (A ∈ (dom R ∪ ran R) → ARA)

Syntax hints:   → wi 2   ∨ wo 195  ∃wex 678   = wceq 1091   ∈ wcel 1092   ∪ 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

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