Theorem issref 5007

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 2460	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥𝑅𝑥))
2		vex 2740	. . . . 5 ⊢ 𝑥 ∈ V
3		opelresi 4914	. . . . 5 ⊢ (𝑥 ∈ V → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ 𝑥 ∈ 𝐴))
4	2, 3	ax-mp 5	. . . 4 ⊢ (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) ↔ 𝑥 ∈ 𝐴)
5		df-br 4001	. . . . 5 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
6	5	bicomi 132	. . . 4 ⊢ (⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ 𝑥𝑅𝑥)
7	4, 6	imbi12i 239	. . 3 ⊢ ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ (𝑥 ∈ 𝐴 → 𝑥𝑅𝑥))
8	7	albii 1470	. 2 ⊢ (∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥𝑅𝑥))
9		ralidm 3523	. . . . . 6 ⊢ (∀𝑥 ∈ V ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
10		ralv 2754	. . . . . 6 ⊢ (∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
11	9, 10	bitri 184	. . . . 5 ⊢ (∀𝑥 ∈ V ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
12		df-ral 2460	. . . . . . . . 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 4910	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑧⟩ ∈ I ∧ 𝑥 ∈ 𝐴)))
15		df-br 4001	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 I 𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ I )
16		vex 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ V
17	16	ideq 4775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 I 𝑧 ↔ 𝑥 = 𝑧)
18		opelresi 4914	. . . . . . . . . . . . . . . . . . . . 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 3777	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑧⟩)
21	20	eleq1d 2246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝑅))
22	21	biimpcd 159	. . . . . . . . . . . . . . . . . . . . . 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 164	. . . . . . . . . . . . . . . . . . . 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 121	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 I 𝑧 → (𝑥 ∈ 𝐴 → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
28	15, 27	sylbir 135	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑧⟩ ∈ I → (𝑥 ∈ 𝐴 → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
29	28	imp 124	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑥, 𝑧⟩ ∈ I ∧ 𝑥 ∈ 𝐴) → ((⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
30	14, 29	syl6bi 163	. . . . . . . . . . . . . 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 2549	. . . . . . . . . . . 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 5	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅)) → ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
35	34	sps 1537	. . . . . . . . 9 ⊢ (∀𝑥(𝑥 ∈ V → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅)) → ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
36	12, 35	sylbi 121	. . . . . . . 8 ⊢ (∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
37	36	ralimi 2540	. . . . . . 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 2240	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → (𝑦 ∈ ( I ↾ 𝐴) ↔ ⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴)))
39		eleq1 2240	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → (𝑦 ∈ 𝑅 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝑅))
40	38, 39	imbi12d 234	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
41	40	ralxp 4766	. . . . . . 7 ⊢ (∀𝑦 ∈ (V × V)(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ V ∀𝑧 ∈ V (⟨𝑥, 𝑧⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
42	37, 41	sylibr 134	. . . . . 6 ⊢ (∀𝑥 ∈ V ∀𝑥 ∈ V (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ∀𝑦 ∈ (V × V)(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
43		df-ral 2460	. . . . . . 7 ⊢ (∀𝑦 ∈ (V × V)(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑦(𝑦 ∈ (V × V) → (𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)))
44		relres 4931	. . . . . . . . . . . 12 ⊢ Rel ( I ↾ 𝐴)
45		df-rel 4630	. . . . . . . . . . . 12 ⊢ (Rel ( I ↾ 𝐴) ↔ ( I ↾ 𝐴) ⊆ (V × V))
46	44, 45	mpbi 145	. . . . . . . . . . 11 ⊢ ( I ↾ 𝐴) ⊆ (V × V)
47	46	sseli 3151	. . . . . . . . . 10 ⊢ (𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ (V × V))
48	47	ancri 324	. . . . . . . . 9 ⊢ (𝑦 ∈ ( I ↾ 𝐴) → (𝑦 ∈ (V × V) ∧ 𝑦 ∈ ( I ↾ 𝐴)))
49		pm3.31 262	. . . . . . . . 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 1455	. . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ (V × V) → (𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)) → ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
52	43, 51	sylbi 121	. . . . . 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 135	. . . 4 ⊢ (∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
55		dfss2 3144	. . . 4 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
56	54, 55	sylibr 134	. . 3 ⊢ (∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) → ( I ↾ 𝐴) ⊆ 𝑅)
57		ssel 3149	. . . 4 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 → (⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
58	57	alrimiv 1874	. . 3 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 → ∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅))
59	56, 58	impbii 126	. 2 ⊢ (∀𝑥(⟨𝑥, 𝑥⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑥⟩ ∈ 𝑅) ↔ ( I ↾ 𝐴) ⊆ 𝑅)
60	1, 8, 59	3bitr2ri 209	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2737   ⊆ wss 3129  ⟨cop 3594   class class class wbr 4000   I cid 4285   × cxp 4621   ↾ cres 4625  Rel wrel 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-iun 3886  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-res 4635

This theorem is referenced by: (None)