Theorem fiint 4703

Description: Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of A is in A." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally.

Assertion

Ref	Expression
fiint	|- (A.x e. A A.y e. A (x i^i y) e. A <-> A.x((x (_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A))

Distinct variable group:   x,y,A

Proof of Theorem fiint

Step	Hyp	Ref	Expression
1		isfi 4523	. . . . . . 7 |- (x e. Fin <-> E.y e. om x ~~ y)
2		breq1 2695	. . . . . . . . . . . . . . . 16 |- (y = (/) -> (y ~~ x <-> (/) ~~ x))
3	2	anbi2d 619	. . . . . . . . . . . . . . 15 |- (y = (/) -> (((x (_ A /\ x =/= (/)) /\ y ~~ x) <-> ((x (_ A /\ x =/= (/)) /\ (/) ~~ x)))
4	3	imbi1d 616	. . . . . . . . . . . . . 14 |- (y = (/) -> ((((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A) <-> (((x (_ A /\ x =/= (/)) /\ (/) ~~ x) -> |^|x e. A)))
5	4	albidv 1316	. . . . . . . . . . . . 13 |- (y = (/) -> (A.x(((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A) <-> A.x(((x (_ A /\ x =/= (/)) /\ (/) ~~ x) -> |^|x e. A)))
6		breq1 2695	. . . . . . . . . . . . . . . 16 |- (y = v -> (y ~~ x <-> v ~~ x))
7	6	anbi2d 619	. . . . . . . . . . . . . . 15 |- (y = v -> (((x (_ A /\ x =/= (/)) /\ y ~~ x) <-> ((x (_ A /\ x =/= (/)) /\ v ~~ x)))
8	7	imbi1d 616	. . . . . . . . . . . . . 14 |- (y = v -> ((((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A) <-> (((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A)))
9	8	albidv 1316	. . . . . . . . . . . . 13 |- (y = v -> (A.x(((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A) <-> A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A)))
10		breq1 2695	. . . . . . . . . . . . . . . 16 |- (y = suc v -> (y ~~ x <-> suc v ~~ x))
11	10	anbi2d 619	. . . . . . . . . . . . . . 15 |- (y = suc v -> (((x (_ A /\ x =/= (/)) /\ y ~~ x) <-> ((x (_ A /\ x =/= (/)) /\ suc v ~~ x)))
12	11	imbi1d 616	. . . . . . . . . . . . . 14 |- (y = suc v -> ((((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A) <-> (((x (_ A /\ x =/= (/)) /\ suc v ~~ x) -> |^|x e. A)))
13	12	albidv 1316	. . . . . . . . . . . . 13 |- (y = suc v -> (A.x(((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A) <-> A.x(((x (_ A /\ x =/= (/)) /\ suc v ~~ x) -> |^|x e. A)))
14		visset 1859	. . . . . . . . . . . . . . . . . . . . 21 |- x e. V
15	14	ensym 4553	. . . . . . . . . . . . . . . . . . . 20 |- ((/) ~~ x -> x ~~ (/))
16		en0 4564	. . . . . . . . . . . . . . . . . . . 20 |- (x ~~ (/) <-> x = (/))
17	15, 16	sylib 196	. . . . . . . . . . . . . . . . . . 19 |- ((/) ~~ x -> x = (/))
18	17	anim1i 332	. . . . . . . . . . . . . . . . . 18 |- (((/) ~~ x /\ x =/= (/)) -> (x = (/) /\ x =/= (/)))
19	18	ancoms 438	. . . . . . . . . . . . . . . . 17 |- ((x =/= (/) /\ (/) ~~ x) -> (x = (/) /\ x =/= (/)))
20	19	adantll 392	. . . . . . . . . . . . . . . 16 |- (((x (_ A /\ x =/= (/)) /\ (/) ~~ x) -> (x = (/) /\ x =/= (/)))
21		pm3.24 661	. . . . . . . . . . . . . . . . . 18 |- -. (x = (/) /\ -. x = (/))
22	21	pm2.21i 77	. . . . . . . . . . . . . . . . 17 |- ((x = (/) /\ -. x = (/)) -> |^|x e. A)
23		df-ne 1630	. . . . . . . . . . . . . . . . 17 |- (x =/= (/) <-> -. x = (/))
24	22, 23	sylan2b 454	. . . . . . . . . . . . . . . 16 |- ((x = (/) /\ x =/= (/)) -> |^|x e. A)
25	20, 24	syl 10	. . . . . . . . . . . . . . 15 |- (((x (_ A /\ x =/= (/)) /\ (/) ~~ x) -> |^|x e. A)
26	25	ax-gen 999	. . . . . . . . . . . . . 14 |- A.x(((x (_ A /\ x =/= (/)) /\ (/) ~~ x) -> |^|x e. A)
27	26	a1i 8	. . . . . . . . . . . . 13 |- (A.z e. A A.w e. A (z i^i w) e. A -> A.x(((x (_ A /\ x =/= (/)) /\ (/) ~~ x) -> |^|x e. A))
28		ax-17 1007	. . . . . . . . . . . . . . 15 |- (A.z e. A A.w e. A (z i^i w) e. A -> A.xA.z e. A A.w e. A (z i^i w) e. A)
29		hba1 1039	. . . . . . . . . . . . . . 15 |- (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> A.xA.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A))
30		ssel 2115	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x (_ A -> ((f` v) e. x -> (f` v) e. A))
31		f1ofo 3803	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (f:suc v-1-1-onto->x -> f:suc v-onto->x)
32		fof 3779	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (f:suc v-onto->x -> f:suc v-->x)
33		visset 1859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- v e. V
34	33	sucid 3051	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- v e. suc v
35		ffvelrn 3928	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((f:suc v-->x /\ v e. suc v) -> (f` v) e. x)
36	34, 35	mpan2 700	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (f:suc v-->x -> (f` v) e. x)
37	31, 32, 36	3syl 20	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (f:suc v-1-1-onto->x -> (f` v) e. x)
38	30, 37	syl5 21	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x (_ A -> (f:suc v-1-1-onto->x -> (f` v) e. A))
39	38	imp 348	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((x (_ A /\ f:suc v-1-1-onto->x) -> (f` v) e. A)
40	39	adantr 389	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((x (_ A /\ f:suc v-1-1-onto->x) /\ (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) /\ A.z e. A A.w e. A (z i^i w) e. A)) -> (f` v) e. A)
41		imassrn 3507	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (f"v) (_ ran f
42		dff1o2 3801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- (f:suc v-1-1-onto->x <-> (f Fn suc v /\ Fun `'f /\ ran f = x))
43		3simp3 796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ((f Fn suc v /\ Fun `'f /\ ran f = x) -> ran f = x)
44	42, 43	sylbi 197	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- (f:suc v-1-1-onto->x -> ran f = x)
45	44	sseq2d 2141	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (f:suc v-1-1-onto->x -> ((f"v) (_ ran f <-> (f"v) (_ x))
46	41, 45	mpbii 191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (f:suc v-1-1-onto->x -> (f"v) (_ x)
47		sstr2 2123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ((f"v) (_ x -> (x (_ A -> (f"v) (_ A))
48	46, 47	syl 10	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (f:suc v-1-1-onto->x -> (x (_ A -> (f"v) (_ A))
49	48	anim1d 563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (f:suc v-1-1-onto->x -> ((x (_ A /\ (f"v) =/= (/)) -> ((f"v) (_ A /\ (f"v) =/= (/))))
50		f1of1 3796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (f:suc v-1-1-onto->x -> f:suc v-1-1->x)
51		sssucid 3050	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- v (_ suc v
52		f1ores 3811	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ((f:suc v-1-1->x /\ v (_ suc v) -> (f |` v):v-1-1-onto->(f"v))
53	51, 52	mpan2 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (f:suc v-1-1->x -> (f |` v):v-1-1-onto->(f"v))
54	33	f1oen 4539	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ((f |` v):v-1-1-onto->(f"v) -> v ~~ (f"v))
55	50, 53, 54	3syl 20	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (f:suc v-1-1-onto->x -> v ~~ (f"v))
56	49, 55	jctird 605	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (f:suc v-1-1-onto->x -> ((x (_ A /\ (f"v) =/= (/)) -> (((f"v) (_ A /\ (f"v) =/= (/)) /\ v ~~ (f"v))))
57		visset 1859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- f e. V
58		imaexg 3508	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (f e. V -> (f"v) e. V)
59	57, 58	ax-mp 7	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (f"v) e. V
60		sseq1 2134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (x = (f"v) -> (x (_ A <-> (f"v) (_ A))
61		neeq1 1633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (x = (f"v) -> (x =/= (/) <-> (f"v) =/= (/)))
62	60, 61	anbi12d 631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (x = (f"v) -> ((x (_ A /\ x =/= (/)) <-> ((f"v) (_ A /\ (f"v) =/= (/))))
63		breq2 2696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (x = (f"v) -> (v ~~ x <-> v ~~ (f"v)))
64	62, 63	anbi12d 631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (x = (f"v) -> (((x (_ A /\ x =/= (/)) /\ v ~~ x) <-> (((f"v) (_ A /\ (f"v) =/= (/)) /\ v ~~ (f"v))))
65		inteq 2603	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (x = (f"v) -> |^|x = |^|(f"v))
66	65	eleq1d 1583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (x = (f"v) -> (|^|x e. A <-> |^|(f"v) e. A))
67	64, 66	imbi12d 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (x = (f"v) -> ((((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) <-> ((((f"v) (_ A /\ (f"v) =/= (/)) /\ v ~~ (f"v)) -> |^|(f"v) e. A)))
68	59, 67	cla4v 1914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> ((((f"v) (_ A /\ (f"v) =/= (/)) /\ v ~~ (f"v)) -> |^|(f"v) e. A))
69	56, 68	sylan9 470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((f:suc v-1-1-onto->x /\ A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A)) -> ((x (_ A /\ (f"v) =/= (/)) -> |^|(f"v) e. A))
70		ineq1 2262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (z = |^|(f"v) -> (z i^i w) = (|^|(f"v) i^i w))
71	70	eleq1d 1583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (z = |^|(f"v) -> ((z i^i w) e. A <-> (|^|(f"v) i^i w) e. A))
72		ineq2 2263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (w = (f` v) -> (|^|(f"v) i^i w) = (|^|(f"v) i^i (f` v)))
73	72	eleq1d 1583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (w = (f` v) -> ((|^|(f"v) i^i w) e. A <-> (|^|(f"v) i^i (f` v)) e. A))
74	71, 73	rcla42v 1926	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((|^|(f"v) e. A /\ (f` v) e. A) -> (A.z e. A A.w e. A (z i^i w) e. A -> (|^|(f"v) i^i (f` v)) e. A))
75	74	ex 371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (|^|(f"v) e. A -> ((f` v) e. A -> (A.z e. A A.w e. A (z i^i w) e. A -> (|^|(f"v) i^i (f` v)) e. A)))
76	69, 75	syl6 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((f:suc v-1-1-onto->x /\ A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A)) -> ((x (_ A /\ (f"v) =/= (/)) -> ((f` v) e. A -> (A.z e. A A.w e. A (z i^i w) e. A -> (|^|(f"v) i^i (f` v)) e. A))))
77	76	com4r 41	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (A.z e. A A.w e. A (z i^i w) e. A -> ((f:suc v-1-1-onto->x /\ A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A)) -> ((x (_ A /\ (f"v) =/= (/)) -> ((f` v) e. A -> (|^|(f"v) i^i (f` v)) e. A))))
78	77	exp4a 378	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (A.z e. A A.w e. A (z i^i w) e. A -> ((f:suc v-1-1-onto->x /\ A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A)) -> (x (_ A -> ((f"v) =/= (/) -> ((f` v) e. A -> (|^|(f"v) i^i (f` v)) e. A)))))
79	78	exp3a 374	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (A.z e. A A.w e. A (z i^i w) e. A -> (f:suc v-1-1-onto->x -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> (x (_ A -> ((f"v) =/= (/) -> ((f` v) e. A -> (|^|(f"v) i^i (f` v)) e. A))))))
80	79	com14 38	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x (_ A -> (f:suc v-1-1-onto->x -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> (A.z e. A A.w e. A (z i^i w) e. A -> ((f"v) =/= (/) -> ((f` v) e. A -> (|^|(f"v) i^i (f` v)) e. A))))))
81	80	imp43 368	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((x (_ A /\ f:suc v-1-1-onto->x) /\ (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) /\ A.z e. A A.w e. A (z i^i w) e. A)) -> ((f"v) =/= (/) -> ((f` v) e. A -> (|^|(f"v) i^i (f` v)) e. A)))
82		df-ne 1630	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((f"v) =/= (/) <-> -. (f"v) = (/))
83	81, 82	syl5ibr 205	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((x (_ A /\ f:suc v-1-1-onto->x) /\ (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) /\ A.z e. A A.w e. A (z i^i w) e. A)) -> (-. (f"v) = (/) -> ((f` v) e. A -> (|^|(f"v) i^i (f` v)) e. A)))
84		inteq 2603	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((f"v) = (/) -> |^|(f"v) = |^|(/))
85		int0 2614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- |^|(/) = V
86	84, 85	syl6eq 1566	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((f"v) = (/) -> |^|(f"v) = V)
87	86	ineq1d 2268	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((f"v) = (/) -> (|^|(f"v) i^i (f` v)) = (V i^i (f` v)))
88		ssv 2133	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (f` v) (_ V
89		sseqin2 2281	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((f` v) (_ V <-> (V i^i (f` v)) = (f` v))
90	88, 89	mpbi 187	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (V i^i (f` v)) = (f` v)
91	87, 90	syl6eq 1566	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((f"v) = (/) -> (|^|(f"v) i^i (f` v)) = (f` v))
92	91	eleq1d 1583	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((f"v) = (/) -> ((|^|(f"v) i^i (f` v)) e. A <-> (f` v) e. A))
93	92	biimprd 152	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((f"v) = (/) -> ((f` v) e. A -> (|^|(f"v) i^i (f` v)) e. A))
94	83, 93	pm2.61d2 127	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((x (_ A /\ f:suc v-1-1-onto->x) /\ (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) /\ A.z e. A A.w e. A (z i^i w) e. A)) -> ((f` v) e. A -> (|^|(f"v) i^i (f` v)) e. A))
95	40, 94	mpd 26	. . . . . . . . . . . . . . . . . . . . . 22 |- (((x (_ A /\ f:suc v-1-1-onto->x) /\ (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) /\ A.z e. A A.w e. A (z i^i w) e. A)) -> (|^|(f"v) i^i (f` v)) e. A)
96		f1ofn 3798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (f:suc v-1-1-onto->x -> f Fn suc v)
97		fnsnfv 3878	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((f Fn suc v /\ v e. suc v) -> {(f` v)} = (f"{v}))
98	34, 97	mpan2 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (f Fn suc v -> {(f` v)} = (f"{v}))
99	96, 98	syl 10	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (f:suc v-1-1-onto->x -> {(f` v)} = (f"{v}))
100	99	uneq2d 2236	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (f:suc v-1-1-onto->x -> ((f"v) u. {(f` v)}) = ((f"v) u. (f"{v})))
101		df-suc 2981	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- suc v = (v u. {v})
102	101	imaeq2i 3494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (f"suc v) = (f"(v u. {v}))
103		imaundi 3545	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (f"(v u. {v})) = ((f"v) u. (f"{v}))
104	102, 103	eqtr2i 1539	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((f"v) u. (f"{v})) = (f"suc v)
105	100, 104	syl6eq 1566	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (f:suc v-1-1-onto->x -> ((f"v) u. {(f` v)}) = (f"suc v))
106		foima 3784	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (f:suc v-onto->x -> (f"suc v) = x)
107	31, 106	syl 10	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (f:suc v-1-1-onto->x -> (f"suc v) = x)
108	105, 107	eqtrd 1550	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (f:suc v-1-1-onto->x -> ((f"v) u. {(f` v)}) = x)
109	108	inteqd 2605	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (f:suc v-1-1-onto->x -> |^|((f"v) u. {(f` v)}) = |^|x)
110		fvex 3843	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (f` v) e. V
111	110	intunsn 2632	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- |^|((f"v) u. {(f` v)}) = (|^|(f"v) i^i (f` v))
112	109, 111	syl5eqr 1564	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (f:suc v-1-1-onto->x -> (|^|(f"v) i^i (f` v)) = |^|x)
113	112	eleq1d 1583	. . . . . . . . . . . . . . . . . . . . . . 23 |- (f:suc v-1-1-onto->x -> ((|^|(f"v) i^i (f` v)) e. A <-> |^|x e. A))
114	113	ad2antlr 405	. . . . . . . . . . . . . . . . . . . . . 22 |- (((x (_ A /\ f:suc v-1-1-onto->x) /\ (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) /\ A.z e. A A.w e. A (z i^i w) e. A)) -> ((|^|(f"v) i^i (f` v)) e. A <-> |^|x e. A))
115	95, 114	mpbid 193	. . . . . . . . . . . . . . . . . . . . 21 |- (((x (_ A /\ f:suc v-1-1-onto->x) /\ (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) /\ A.z e. A A.w e. A (z i^i w) e. A)) -> |^|x e. A)
116	115	exp43 384	. . . . . . . . . . . . . . . . . . . 20 |- (x (_ A -> (f:suc v-1-1-onto->x -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> (A.z e. A A.w e. A (z i^i w) e. A -> |^|x e. A))))
117	116	19.23adv 1251	. . . . . . . . . . . . . . . . . . 19 |- (x (_ A -> (E.f f:suc v-1-1-onto->x -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> (A.z e. A A.w e. A (z i^i w) e. A -> |^|x e. A))))
118	14	bren 4518	. . . . . . . . . . . . . . . . . . 19 |- (suc v ~~ x <-> E.f f:suc v-1-1-onto->x)
119	117, 118	syl5ib 204	. . . . . . . . . . . . . . . . . 18 |- (x (_ A -> (suc v ~~ x -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> (A.z e. A A.w e. A (z i^i w) e. A -> |^|x e. A))))
120	119	imp 348	. . . . . . . . . . . . . . . . 17 |- ((x (_ A /\ suc v ~~ x) -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> (A.z e. A A.w e. A (z i^i w) e. A -> |^|x e. A)))
121	120	adantlr 393	. . . . . . . . . . . . . . . 16 |- (((x (_ A /\ x =/= (/)) /\ suc v ~~ x) -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> (A.z e. A A.w e. A (z i^i w) e. A -> |^|x e. A)))
122	121	com13 33	. . . . . . . . . . . . . . 15 |- (A.z e. A A.w e. A (z i^i w) e. A -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> (((x (_ A /\ x =/= (/)) /\ suc v ~~ x) -> |^|x e. A)))
123	28, 29, 122	19.21ad 1095	. . . . . . . . . . . . . 14 |- (A.z e. A A.w e. A (z i^i w) e. A -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> A.x(((x (_ A /\ x =/= (/)) /\ suc v ~~ x) -> |^|x e. A)))
124	123	a1i 8	. . . . . . . . . . . . 13 |- (v e. om -> (A.z e. A A.w e. A (z i^i w) e. A -> (A.x(((x (_ A /\ x =/= (/)) /\ v ~~ x) -> |^|x e. A) -> A.x(((x (_ A /\ x =/= (/)) /\ suc v ~~ x) -> |^|x e. A))))
125	5, 9, 13, 27, 124	finds2 3246	. . . . . . . . . . . 12 |- (y e. om -> (A.z e. A A.w e. A (z i^i w) e. A -> A.x(((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A)))
126		ax-4 1009	. . . . . . . . . . . 12 |- (A.x(((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A) -> (((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A))
127	125, 126	syl6 22	. . . . . . . . . . 11 |- (y e. om -> (A.z e. A A.w e. A (z i^i w) e. A -> (((x (_ A /\ x =/= (/)) /\ y ~~ x) -> |^|x e. A)))
128	127	exp4a 378	. . . . . . . . . 10 |- (y e. om -> (A.z e. A A.w e. A (z i^i w) e. A -> ((x (_ A /\ x =/= (/)) -> (y ~~ x -> |^|x e. A))))
129	128	com24 37	. . . . . . . . 9 |- (y e. om -> (y ~~ x -> ((x (_ A /\ x =/= (/)) -> (A.z e. A A.w e. A (z i^i w) e. A -> |^|x e. A))))
130		visset 1859	. . . . . . . . . 10 |- y e. V
131	130	ensym 4553	. . . . . . . . 9 |- (x ~~ y -> y ~~ x)
132	129, 131	syl5 21	. . . . . . . 8 |- (y e. om -> (x ~~ y -> ((x (_ A /\ x =/= (/)) -> (A.z e. A A.w e. A (z i^i w) e. A -> |^|x e. A))))
133	132	r19.23aiv 1789	. . . . . . 7 |- (E.y e. om x ~~ y -> ((x (_ A /\ x =/= (/)) -> (A.z e. A A.w e. A (z i^i w) e. A -> |^|x e. A)))
134	1, 133	sylbi 197	. . . . . 6 |- (x e. Fin -> ((x (_ A /\ x =/= (/)) -> (A.z e. A A.w e. A (z i^i w) e. A -> |^|x e. A)))
135	134	com13 33	. . . . 5 |- (A.z e. A A.w e. A (z i^i w) e. A -> ((x (_ A /\ x =/= (/)) -> (x e. Fin -> |^|x e. A)))
136	135	imp3a 359	. . . 4 |- (A.z e. A A.w e. A (z i^i w) e. A -> (((x (_ A /\ x =/= (/)) /\ x e. Fin) -> |^|x e. A))
137	136	19.21aiv 1324	. . 3 |- (A.z e. A A.w e. A (z i^i w) e. A -> A.x(((x (_ A /\ x =/= (/)) /\ x e. Fin) -> |^|x e. A))
138		zfpair2 2856	. . . . . 6 |- {z, w} e. V
139		sseq1 2134	. . . . . . . . 9 |- (x = {z, w} -> (x (_ A <-> {z, w} (_ A))
140		neeq1 1633	. . . . . . . . 9 |- (x = {z, w} -> (x =/= (/) <-> {z, w} =/= (/)))
141	139, 140	anbi12d 631	. . . . . . . 8 |- (x = {z, w} -> ((x (_ A /\ x =/= (/)) <-> ({z, w} (_ A /\ {z, w} =/= (/))))
142		eleq1 1577	. . . . . . . 8 |- (x = {z, w} -> (x e. Fin <-> {z, w} e. Fin))
143	141, 142	anbi12d 631	. . . . . . 7 |- (x = {z, w} -> (((x (_ A /\ x =/= (/)) /\ x e. Fin) <-> (({z, w} (_ A /\ {z, w} =/= (/)) /\ {z, w} e. Fin)))
144		inteq 2603	. . . . . . . 8 |- (x = {z, w} -> |^|x = |^|{z, w})
145	144	eleq1d 1583	. . . . . . 7 |- (x = {z, w} -> (|^|x e. A <-> |^|{z, w} e. A))
146	143, 145	imbi12d 629	. . . . . 6 |- (x = {z, w} -> ((((x (_ A /\ x =/= (/)) /\ x e. Fin) -> |^|x e. A) <-> ((({z, w} (_ A /\ {z, w} =/= (/)) /\ {z, w} e. Fin) -> |^|{z, w} e. A)))
147	138, 146	cla4v 1914	. . . . 5 |- (A.x(((x (_ A /\ x =/= (/)) /\ x e. Fin) -> |^|x e. A) -> ((({z, w} (_ A /\ {z, w} =/= (/)) /\ {z, w} e. Fin) -> |^|{z, w} e. A))
148		visset 1859	. . . . . . 7 |- z e. V
149		visset 1859	. . . . . . 7 |- w e. V
150	148, 149	prss 2536	. . . . . 6 |- ((z e. A /\ w e. A) <-> {z, w} (_ A)
151	148	prnz 2523	. . . . . . 7 |- {z, w} =/= (/)
152	151	biantru 729	. . . . . 6 |- ({z, w} (_ A <-> ({z, w} (_ A /\ {z, w} =/= (/)))
153		prfi 4700	. . . . . . 7 |- {z, w} e. Fin
154	153	biantru 729	. . . . . 6 |- (({z, w} (_ A /\ {z, w} =/= (/)) <-> (({z, w} (_ A /\ {z, w} =/= (/)) /\ {z, w} e. Fin))
155	150, 152, 154	3bitrri 176	. . . . 5 |- ((({z, w} (_ A /\ {z, w} =/= (/)) /\ {z, w} e. Fin) <-> (z e. A /\ w e. A))
156	148, 149	intpr 2630	. . . . . 6 |- |^|{z, w} = (z i^i w)
157	156	eleq1i 1580	. . . . 5 |- (|^|{z, w} e. A <-> (z i^i w) e. A)
158	147, 155, 157	3imtr3g 555	. . . 4 |- (A.x(((x (_ A /\ x =/= (/)) /\ x e. Fin) -> |^|x e. A) -> ((z e. A /\ w e. A) -> (z i^i w) e. A))
159	158	r19.21aivv 1766	. . 3 |- (A.x(((x (_ A /\ x =/= (/)) /\ x e. Fin) -> |^|x e. A) -> A.z e. A A.w e. A (z i^i w) e. A)
160	137, 159	impbii 155	. 2 |- (A.z e. A A.w e. A (z i^i w) e. A <-> A.x(((x (_ A /\ x =/= (/)) /\ x e. Fin) -> |^|x e. A))
161		ineq1 2262	. . . 4 |- (x = z -> (x i^i y) = (z i^i y))
162	161	eleq1d 1583	. . 3 |- (x = z -> ((x i^i y) e. A <-> (z i^i y) e. A))
163		ineq2 2263	. . . 4 |- (y = w -> (z i^i y) = (z i^i w))
164	163	eleq1d 1583	. . 3 |- (y = w -> ((z i^i y) e. A <-> (z i^i w) e. A))
165	162, 164	cbvral2v 1849	. 2 |- (A.x e. A A.y e. A (x i^i y) e. A <-> A.z e. A A.w e. A (z i^i w) e. A)
166		df-3an 783	. . . 4 |- ((x (_ A /\ x =/= (/) /\ x e. Fin) <-> ((x (_ A /\ x =/= (/)) /\ x e. Fin))
167	166	imbi1i 184	. . 3 |- (((x (_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) <-> (((x (_ A /\ x =/= (/)) /\ x e. Fin) -> |^|x e. A))
168	167	albii 1035	. 2 |- (A.x((x (_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) <-> A.x(((x (_ A /\ x =/= (/)) /\ x e. Fin) -> |^|x e. A))
169	160, 165, 168	3bitr4i 181	1 |- (A.x e. A A.y e. A (x i^i y) e. A <-> A.x((x (_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221   /\ w3a 781  A.wal 990   = wceq 992   e. wcel 994  E.wex 1016   =/= wne 1628  A.wral 1691  E.wrex 1692  Vcvv 1857   u. cun 2097   i^i cin 2098   (_ wss 2099  (/)c0 2332  {csn 2467  {cpr 2468  |^|cint 2600   class class class wbr 2692  suc csuc 2977  omcom 3218  `'ccnv 3250  ran crn 3252   |` cres 3253  "cima 3254  Fun wfun 3257   Fn wfn 3258  -->wf 3259  -1-1->wf1 3260  -onto->wfo 3261  -1-1-onto->wf1o 3262  ` cfv 3263   ~~ cen 4505  Fincfn 4508

This theorem is referenced by:  abfii1 4704  abfii4 4707  istop2g 7809  istps4 7821  fipfil2 11077  filint2 11084  inficlALT 11424

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-rdg 4233  df-1o 4269  df-oadd 4271  df-er 4401  df-en 4509  df-fin 4512

Copyright terms: Public domain