Theorem issref 4879

Description: Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)

Assertion

Ref	Expression
issref	⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem issref

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 2395	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥𝑅𝑥))
2		vex 2660	. . . . 5 ⊢ 𝑥 ∈ V
3		opelresi 4788	. . . . 5 ⊢ (𝑥 ∈ V → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ 𝑥 ∈ 𝐴))
4	2, 3	ax-mp 7	. . . 4 ⊢ (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ 𝑥 ∈ 𝐴)
5		df-br 3896	. . . . 5 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
6	5	bicomi 131	. . . 4 ⊢ (⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ 𝑥𝑅𝑥)
7	4, 6	imbi12i 238	. . 3 ⊢ ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ (𝑥 ∈ 𝐴 → 𝑥𝑅𝑥))
8	7	albii 1429	. 2 ⊢ (∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥𝑅𝑥))
9		ralidm 3429	. . . . . 6 ⊢ (∀𝑥 ∈ V ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
10		ralv 2674	. . . . . 6 ⊢ (∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
11	9, 10	bitri 183	. . . . 5 ⊢ (∀𝑥 ∈ V ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
12		df-ral 2395	. . . . . . . . 9 ⊢ (∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥(𝑥 ∈ V → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅)))
13		pm2.27 40	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → ((𝑥 ∈ V → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅)) → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅)))
14		opelresg 4784	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑧⟩ ∈ I ∧ 𝑥 ∈ 𝐴)))
15		df-br 3896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 I 𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ I )
16		vex 2660	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ V
17	16	ideq 4651	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
18		opelresi 4788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝐴 → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ 𝑥 ∈ 𝐴))
19		pm2.27 40	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
20		opeq2 3672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑧⟩)
21	20	eleq1d 2183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝑅))
22	21	biimpcd 158	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑥, 𝑥⟩ ∈ 𝑅 → (𝑥 = 𝑧 → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
23	19, 22	syl6 33	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → (𝑥 = 𝑧 → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
24	18, 23	syl6bir 163	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → (𝑥 = 𝑧 → ⟨𝑥, 𝑧⟩ ∈ 𝑅))))
25	24	pm2.43i 49	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → (𝑥 = 𝑧 → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
26	25	com3r 79	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
27	17, 26	sylbi 120	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 I 𝑧 → (𝑥 ∈ 𝐴 → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
28	15, 27	sylbir 134	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑧⟩ ∈ I → (𝑥 ∈ 𝐴 → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
29	28	imp 123	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑥, 𝑧⟩ ∈ I ∧ 𝑥 ∈ 𝐴) → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
30	14, 29	syl6bi 162	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
31	30	com3r 79	. . . . . . . . . . . . 13 ⊢ ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → (𝑧 ∈ V → (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
32	31	ralrimiv 2478	. . . . . . . . . . . 12 ⊢ ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
33	13, 32	syl6 33	. . . . . . . . . . 11 ⊢ (𝑥 ∈ V → ((𝑥 ∈ V → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅)) → ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
34	2, 33	ax-mp 7	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅)) → ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
35	34	sps 1500	. . . . . . . . 9 ⊢ (∀𝑥(𝑥 ∈ V → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅)) → ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
36	12, 35	sylbi 120	. . . . . . . 8 ⊢ (∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
37	36	ralimi 2469	. . . . . . 7 ⊢ (∀𝑥 ∈ V ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ∀𝑥 ∈ V ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
38		eleq1 2177	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → (𝑦 ∈ ( I ↾ 𝐴) ↔ ⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴)))
39		eleq1 2177	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → (𝑦 ∈ 𝑅 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝑅))
40	38, 39	imbi12d 233	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
41	40	ralxp 4642	. . . . . . 7 ⊢ (∀𝑦 ∈ (V × V)(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ V ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
42	37, 41	sylibr 133	. . . . . 6 ⊢ (∀𝑥 ∈ V ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ∀𝑦 ∈ (V × V)(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
43		df-ral 2395	. . . . . . 7 ⊢ (∀𝑦 ∈ (V × V)(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑦(𝑦 ∈ (V × V) → (𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)))
44		relres 4805	. . . . . . . . . . . 12 ⊢ Rel ( I ↾ 𝐴)
45		df-rel 4506	. . . . . . . . . . . 12 ⊢ (Rel ( I ↾ 𝐴) ↔ ( I ↾ 𝐴) ⊆ (V × V))
46	44, 45	mpbi 144	. . . . . . . . . . 11 ⊢ ( I ↾ 𝐴) ⊆ (V × V)
47	46	sseli 3059	. . . . . . . . . 10 ⊢ (𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ (V × V))
48	47	ancri 320	. . . . . . . . 9 ⊢ (𝑦 ∈ ( I ↾ 𝐴) → (𝑦 ∈ (V × V) ∧ 𝑦 ∈ ( I ↾ 𝐴)))
49		pm3.31 260	. . . . . . . . 9 ⊢ ((𝑦 ∈ (V × V) → (𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)) → ((𝑦 ∈ (V × V) ∧ 𝑦 ∈ ( I ↾ 𝐴)) → 𝑦 ∈ 𝑅))
50	48, 49	syl5 32	. . . . . . . 8 ⊢ ((𝑦 ∈ (V × V) → (𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)) → (𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
51	50	alimi 1414	. . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ (V × V) → (𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)) → ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
52	43, 51	sylbi 120	. . . . . 6 ⊢ (∀𝑦 ∈ (V × V)(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) → ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
53	42, 52	syl 14	. . . . 5 ⊢ (∀𝑥 ∈ V ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
54	11, 53	sylbir 134	. . . 4 ⊢ (∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
55		dfss2 3052	. . . 4 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
56	54, 55	sylibr 133	. . 3 ⊢ (∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ( I ↾ 𝐴) ⊆ 𝑅)
57		ssel 3057	. . . 4 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
58	57	alrimiv 1828	. . 3 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 → ∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
59	56, 58	impbii 125	. 2 ⊢ (∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ( I ↾ 𝐴) ⊆ 𝑅)
60	1, 8, 59	3bitr2ri 208	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1312   = wceq 1314   ∈ wcel 1463  ∀wral 2390  Vcvv 2657   ⊆ wss 3037  ⟨cop 3496   class class class wbr 3895   I cid 4170   × cxp 4497   ↾ cres 4501  Rel wrel 4504

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-iun 3781  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-res 4511

This theorem is referenced by: (None)