Theorem iinerm 6114

Description: The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Assertion

Ref	Expression
iinerm	⊢ ((∃y y ∈ A ∧ ∀x ∈ A 𝑅 Er B) → ∩ x ∈ A 𝑅 Er B)

Distinct variable groups:   x,A   x,B   y,A

Allowed substitution hints:   B(y)   𝑅(x,y)

Proof of Theorem iinerm

Dummy variables u 𝑎 v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2097	. . . 4 ⊢ (x = 𝑎 → (x ∈ A ↔ 𝑎 ∈ A))
2	1	cbvexv 1792	. . 3 ⊢ (∃x x ∈ A ↔ ∃𝑎 𝑎 ∈ A)
3		eleq1 2097	. . . 4 ⊢ (𝑎 = y → (𝑎 ∈ A ↔ y ∈ A))
4	3	cbvexv 1792	. . 3 ⊢ (∃𝑎 𝑎 ∈ A ↔ ∃y y ∈ A)
5	2, 4	bitri 173	. 2 ⊢ (∃x x ∈ A ↔ ∃y y ∈ A)
6		r19.2m 3303	. . . . 5 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → ∃x ∈ A 𝑅 Er B)
7		errel 6051	. . . . . . 7 ⊢ (𝑅 Er B → Rel 𝑅)
8		df-rel 4295	. . . . . . 7 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
9	7, 8	sylib 127	. . . . . 6 ⊢ (𝑅 Er B → 𝑅 ⊆ (V × V))
10	9	reximi 2410	. . . . 5 ⊢ (∃x ∈ A 𝑅 Er B → ∃x ∈ A 𝑅 ⊆ (V × V))
11		iinss 3699	. . . . 5 ⊢ (∃x ∈ A 𝑅 ⊆ (V × V) → ∩ x ∈ A 𝑅 ⊆ (V × V))
12	6, 10, 11	3syl 17	. . . 4 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → ∩ x ∈ A 𝑅 ⊆ (V × V))
13		df-rel 4295	. . . 4 ⊢ (Rel ∩ x ∈ A 𝑅 ↔ ∩ x ∈ A 𝑅 ⊆ (V × V))
14	12, 13	sylibr 137	. . 3 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → Rel ∩ x ∈ A 𝑅)
15		id 19	. . . . . . . . . 10 ⊢ (𝑅 Er B → 𝑅 Er B)
16	15	ersymb 6056	. . . . . . . . 9 ⊢ (𝑅 Er B → (u𝑅v ↔ v𝑅u))
17	16	biimpd 132	. . . . . . . 8 ⊢ (𝑅 Er B → (u𝑅v → v𝑅u))
18		df-br 3756	. . . . . . . 8 ⊢ (u𝑅v ↔ ⟨u, v⟩ ∈ 𝑅)
19		df-br 3756	. . . . . . . 8 ⊢ (v𝑅u ↔ ⟨v, u⟩ ∈ 𝑅)
20	17, 18, 19	3imtr3g 193	. . . . . . 7 ⊢ (𝑅 Er B → (⟨u, v⟩ ∈ 𝑅 → ⟨v, u⟩ ∈ 𝑅))
21	20	ral2imi 2379	. . . . . 6 ⊢ (∀x ∈ A 𝑅 Er B → (∀x ∈ A ⟨u, v⟩ ∈ 𝑅 → ∀x ∈ A ⟨v, u⟩ ∈ 𝑅))
22	21	adantl 262	. . . . 5 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → (∀x ∈ A ⟨u, v⟩ ∈ 𝑅 → ∀x ∈ A ⟨v, u⟩ ∈ 𝑅))
23		df-br 3756	. . . . . 6 ⊢ (u∩ x ∈ A 𝑅v ↔ ⟨u, v⟩ ∈ ∩ x ∈ A 𝑅)
24		vex 2554	. . . . . . . 8 ⊢ u ∈ V
25		vex 2554	. . . . . . . 8 ⊢ v ∈ V
26	24, 25	opex 3957	. . . . . . 7 ⊢ ⟨u, v⟩ ∈ V
27		eliin 3653	. . . . . . 7 ⊢ (⟨u, v⟩ ∈ V → (⟨u, v⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨u, v⟩ ∈ 𝑅))
28	26, 27	ax-mp 7	. . . . . 6 ⊢ (⟨u, v⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨u, v⟩ ∈ 𝑅)
29	23, 28	bitri 173	. . . . 5 ⊢ (u∩ x ∈ A 𝑅v ↔ ∀x ∈ A ⟨u, v⟩ ∈ 𝑅)
30		df-br 3756	. . . . . 6 ⊢ (v∩ x ∈ A 𝑅u ↔ ⟨v, u⟩ ∈ ∩ x ∈ A 𝑅)
31	25, 24	opex 3957	. . . . . . 7 ⊢ ⟨v, u⟩ ∈ V
32		eliin 3653	. . . . . . 7 ⊢ (⟨v, u⟩ ∈ V → (⟨v, u⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨v, u⟩ ∈ 𝑅))
33	31, 32	ax-mp 7	. . . . . 6 ⊢ (⟨v, u⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨v, u⟩ ∈ 𝑅)
34	30, 33	bitri 173	. . . . 5 ⊢ (v∩ x ∈ A 𝑅u ↔ ∀x ∈ A ⟨v, u⟩ ∈ 𝑅)
35	22, 29, 34	3imtr4g 194	. . . 4 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → (u∩ x ∈ A 𝑅v → v∩ x ∈ A 𝑅u))
36	35	imp 115	. . 3 ⊢ (((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) ∧ u∩ x ∈ A 𝑅v) → v∩ x ∈ A 𝑅u)
37		r19.26 2435	. . . . . 6 ⊢ (∀x ∈ A (⟨u, v⟩ ∈ 𝑅 ∧ ⟨v, w⟩ ∈ 𝑅) ↔ (∀x ∈ A ⟨u, v⟩ ∈ 𝑅 ∧ ∀x ∈ A ⟨v, w⟩ ∈ 𝑅))
38	15	ertr 6057	. . . . . . . . 9 ⊢ (𝑅 Er B → ((u𝑅v ∧ v𝑅w) → u𝑅w))
39		df-br 3756	. . . . . . . . . 10 ⊢ (v𝑅w ↔ ⟨v, w⟩ ∈ 𝑅)
40	18, 39	anbi12i 433	. . . . . . . . 9 ⊢ ((u𝑅v ∧ v𝑅w) ↔ (⟨u, v⟩ ∈ 𝑅 ∧ ⟨v, w⟩ ∈ 𝑅))
41		df-br 3756	. . . . . . . . 9 ⊢ (u𝑅w ↔ ⟨u, w⟩ ∈ 𝑅)
42	38, 40, 41	3imtr3g 193	. . . . . . . 8 ⊢ (𝑅 Er B → ((⟨u, v⟩ ∈ 𝑅 ∧ ⟨v, w⟩ ∈ 𝑅) → ⟨u, w⟩ ∈ 𝑅))
43	42	ral2imi 2379	. . . . . . 7 ⊢ (∀x ∈ A 𝑅 Er B → (∀x ∈ A (⟨u, v⟩ ∈ 𝑅 ∧ ⟨v, w⟩ ∈ 𝑅) → ∀x ∈ A ⟨u, w⟩ ∈ 𝑅))
44	43	adantl 262	. . . . . 6 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → (∀x ∈ A (⟨u, v⟩ ∈ 𝑅 ∧ ⟨v, w⟩ ∈ 𝑅) → ∀x ∈ A ⟨u, w⟩ ∈ 𝑅))
45	37, 44	syl5bir 142	. . . . 5 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → ((∀x ∈ A ⟨u, v⟩ ∈ 𝑅 ∧ ∀x ∈ A ⟨v, w⟩ ∈ 𝑅) → ∀x ∈ A ⟨u, w⟩ ∈ 𝑅))
46		df-br 3756	. . . . . . 7 ⊢ (v∩ x ∈ A 𝑅w ↔ ⟨v, w⟩ ∈ ∩ x ∈ A 𝑅)
47		vex 2554	. . . . . . . . 9 ⊢ w ∈ V
48	25, 47	opex 3957	. . . . . . . 8 ⊢ ⟨v, w⟩ ∈ V
49		eliin 3653	. . . . . . . 8 ⊢ (⟨v, w⟩ ∈ V → (⟨v, w⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨v, w⟩ ∈ 𝑅))
50	48, 49	ax-mp 7	. . . . . . 7 ⊢ (⟨v, w⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨v, w⟩ ∈ 𝑅)
51	46, 50	bitri 173	. . . . . 6 ⊢ (v∩ x ∈ A 𝑅w ↔ ∀x ∈ A ⟨v, w⟩ ∈ 𝑅)
52	29, 51	anbi12i 433	. . . . 5 ⊢ ((u∩ x ∈ A 𝑅v ∧ v∩ x ∈ A 𝑅w) ↔ (∀x ∈ A ⟨u, v⟩ ∈ 𝑅 ∧ ∀x ∈ A ⟨v, w⟩ ∈ 𝑅))
53		df-br 3756	. . . . . 6 ⊢ (u∩ x ∈ A 𝑅w ↔ ⟨u, w⟩ ∈ ∩ x ∈ A 𝑅)
54	24, 47	opex 3957	. . . . . . 7 ⊢ ⟨u, w⟩ ∈ V
55		eliin 3653	. . . . . . 7 ⊢ (⟨u, w⟩ ∈ V → (⟨u, w⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨u, w⟩ ∈ 𝑅))
56	54, 55	ax-mp 7	. . . . . 6 ⊢ (⟨u, w⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨u, w⟩ ∈ 𝑅)
57	53, 56	bitri 173	. . . . 5 ⊢ (u∩ x ∈ A 𝑅w ↔ ∀x ∈ A ⟨u, w⟩ ∈ 𝑅)
58	45, 52, 57	3imtr4g 194	. . . 4 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → ((u∩ x ∈ A 𝑅v ∧ v∩ x ∈ A 𝑅w) → u∩ x ∈ A 𝑅w))
59	58	imp 115	. . 3 ⊢ (((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) ∧ (u∩ x ∈ A 𝑅v ∧ v∩ x ∈ A 𝑅w)) → u∩ x ∈ A 𝑅w)
60		simpl 102	. . . . . . . . . . 11 ⊢ ((𝑅 Er B ∧ u ∈ B) → 𝑅 Er B)
61		simpr 103	. . . . . . . . . . 11 ⊢ ((𝑅 Er B ∧ u ∈ B) → u ∈ B)
62	60, 61	erref 6062	. . . . . . . . . 10 ⊢ ((𝑅 Er B ∧ u ∈ B) → u𝑅u)
63		df-br 3756	. . . . . . . . . 10 ⊢ (u𝑅u ↔ ⟨u, u⟩ ∈ 𝑅)
64	62, 63	sylib 127	. . . . . . . . 9 ⊢ ((𝑅 Er B ∧ u ∈ B) → ⟨u, u⟩ ∈ 𝑅)
65	64	expcom 109	. . . . . . . 8 ⊢ (u ∈ B → (𝑅 Er B → ⟨u, u⟩ ∈ 𝑅))
66	65	ralimdv 2382	. . . . . . 7 ⊢ (u ∈ B → (∀x ∈ A 𝑅 Er B → ∀x ∈ A ⟨u, u⟩ ∈ 𝑅))
67	66	com12 27	. . . . . 6 ⊢ (∀x ∈ A 𝑅 Er B → (u ∈ B → ∀x ∈ A ⟨u, u⟩ ∈ 𝑅))
68	67	adantl 262	. . . . 5 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → (u ∈ B → ∀x ∈ A ⟨u, u⟩ ∈ 𝑅))
69		r19.26 2435	. . . . . . 7 ⊢ (∀x ∈ A (𝑅 Er B ∧ ⟨u, u⟩ ∈ 𝑅) ↔ (∀x ∈ A 𝑅 Er B ∧ ∀x ∈ A ⟨u, u⟩ ∈ 𝑅))
70		r19.2m 3303	. . . . . . . . 9 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A (𝑅 Er B ∧ ⟨u, u⟩ ∈ 𝑅)) → ∃x ∈ A (𝑅 Er B ∧ ⟨u, u⟩ ∈ 𝑅))
71	24, 24	opeldm 4481	. . . . . . . . . . 11 ⊢ (⟨u, u⟩ ∈ 𝑅 → u ∈ dom 𝑅)
72		erdm 6052	. . . . . . . . . . . . 13 ⊢ (𝑅 Er B → dom 𝑅 = B)
73	72	eleq2d 2104	. . . . . . . . . . . 12 ⊢ (𝑅 Er B → (u ∈ dom 𝑅 ↔ u ∈ B))
74	73	biimpa 280	. . . . . . . . . . 11 ⊢ ((𝑅 Er B ∧ u ∈ dom 𝑅) → u ∈ B)
75	71, 74	sylan2 270	. . . . . . . . . 10 ⊢ ((𝑅 Er B ∧ ⟨u, u⟩ ∈ 𝑅) → u ∈ B)
76	75	rexlimivw 2423	. . . . . . . . 9 ⊢ (∃x ∈ A (𝑅 Er B ∧ ⟨u, u⟩ ∈ 𝑅) → u ∈ B)
77	70, 76	syl 14	. . . . . . . 8 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A (𝑅 Er B ∧ ⟨u, u⟩ ∈ 𝑅)) → u ∈ B)
78	77	ex 108	. . . . . . 7 ⊢ (∃x x ∈ A → (∀x ∈ A (𝑅 Er B ∧ ⟨u, u⟩ ∈ 𝑅) → u ∈ B))
79	69, 78	syl5bir 142	. . . . . 6 ⊢ (∃x x ∈ A → ((∀x ∈ A 𝑅 Er B ∧ ∀x ∈ A ⟨u, u⟩ ∈ 𝑅) → u ∈ B))
80	79	expdimp 246	. . . . 5 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → (∀x ∈ A ⟨u, u⟩ ∈ 𝑅 → u ∈ B))
81	68, 80	impbid 120	. . . 4 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → (u ∈ B ↔ ∀x ∈ A ⟨u, u⟩ ∈ 𝑅))
82		df-br 3756	. . . . 5 ⊢ (u∩ x ∈ A 𝑅u ↔ ⟨u, u⟩ ∈ ∩ x ∈ A 𝑅)
83	24, 24	opex 3957	. . . . . 6 ⊢ ⟨u, u⟩ ∈ V
84		eliin 3653	. . . . . 6 ⊢ (⟨u, u⟩ ∈ V → (⟨u, u⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨u, u⟩ ∈ 𝑅))
85	83, 84	ax-mp 7	. . . . 5 ⊢ (⟨u, u⟩ ∈ ∩ x ∈ A 𝑅 ↔ ∀x ∈ A ⟨u, u⟩ ∈ 𝑅)
86	82, 85	bitri 173	. . . 4 ⊢ (u∩ x ∈ A 𝑅u ↔ ∀x ∈ A ⟨u, u⟩ ∈ 𝑅)
87	81, 86	syl6bbr 187	. . 3 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → (u ∈ B ↔ u∩ x ∈ A 𝑅u))
88	14, 36, 59, 87	iserd 6068	. 2 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A 𝑅 Er B) → ∩ x ∈ A 𝑅 Er B)
89	5, 88	sylanbr 269	1 ⊢ ((∃y y ∈ A ∧ ∀x ∈ A 𝑅 Er B) → ∩ x ∈ A 𝑅 Er B)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  Vcvv 2551   ⊆ wss 2911  ⟨cop 3370  ∩ ciin 3649   class class class wbr 3755   × cxp 4286  dom cdm 4288  Rel wrel 4293   Er wer 6039

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iin 3651  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042

This theorem is referenced by:  riinerm  6115