Theorem locfinref 29908

Description: A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function f from the original cover U, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)

Hypotheses

Ref	Expression
locfinref.x	|- X = U. J
locfinref.1	|- ( ph -> U C_ J )
locfinref.2	|- ( ph -> X = U. U )
locfinref.3	|- ( ph -> V C_ J )
locfinref.4	|- ( ph -> V Ref U )
locfinref.5	|- ( ph -> V e. ( LocFin ` J ) )

Assertion

Ref	Expression
locfinref	|- ( ph -> E. f ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) )

Distinct variable groups:   f, J   U, f   f, V   ph, f

Allowed substitution hint:   X( f)

Proof of Theorem locfinref

Dummy variables g x n s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f0 6086	. . . 4 |- (/) : (/) --> J
2		simpr 477	. . . . 5 |- ( ( ph /\ U = (/) ) -> U = (/) )
3	2	feq2d 6031	. . . 4 |- ( ( ph /\ U = (/) ) -> ( (/) : U --> J <-> (/) : (/) --> J ) )
4	1, 3	mpbiri 248	. . 3 |- ( ( ph /\ U = (/) ) -> (/) : U --> J )
5		rn0 5377	. . . . 5 |- ran (/) = (/)
6		0ex 4790	. . . . . 6 |- (/) e. _V
7		refref 21316	. . . . . 6 |- ( (/) e. _V -> (/) Ref (/) )
8	6, 7	ax-mp 5	. . . . 5 |- (/) Ref (/)
9	5, 8	eqbrtri 4674	. . . 4 |- ran (/) Ref (/)
10	9, 2	syl5breqr 4691	. . 3 |- ( ( ph /\ U = (/) ) -> ran (/) Ref U )
11		sn0top 20803	. . . . . 6 |- { (/) } e. Top
12	11	a1i 11	. . . . 5 |- ( ( ph /\ U = (/) ) -> { (/) } e. Top )
13		eqidd 2623	. . . . 5 |- ( ( ph /\ U = (/) ) -> (/) = (/) )
14		ral0 4076	. . . . . 6 |- A. x e. (/) E. n e. { (/) } ( x e. n /\ { s e. ran (/) | ( s i^i n ) =/= (/) } e. Fin )
15	14	a1i 11	. . . . 5 |- ( ( ph /\ U = (/) ) -> A. x e. (/) E. n e. { (/) } ( x e. n /\ { s e. ran (/) | ( s i^i n ) =/= (/) } e. Fin ) )
16	6	unisn 4451	. . . . . . 7 |- U. { (/) } = (/)
17	16	eqcomi 2631	. . . . . 6 |- (/) = U. { (/) }
18	5	unieqi 4445	. . . . . . 7 |- U. ran (/) = U. (/)
19		uni0 4465	. . . . . . 7 |- U. (/) = (/)
20	18, 19	eqtr2i 2645	. . . . . 6 |- (/) = U. ran (/)
21	17, 20	islocfin 21320	. . . . 5 |- ( ran (/) e. ( LocFin ` { (/) } ) <-> ( { (/) } e. Top /\ (/) = (/) /\ A. x e. (/) E. n e. { (/) } ( x e. n /\ { s e. ran (/) | ( s i^i n ) =/= (/) } e. Fin ) ) )
22	12, 13, 15, 21	syl3anbrc 1246	. . . 4 |- ( ( ph /\ U = (/) ) -> ran (/) e. ( LocFin ` { (/) } ) )
23		locfinref.2	. . . . . . . . 9 |- ( ph -> X = U. U )
24	23	adantr 481	. . . . . . . 8 |- ( ( ph /\ U = (/) ) -> X = U. U )
25	2	unieqd 4446	. . . . . . . 8 |- ( ( ph /\ U = (/) ) -> U. U = U. (/) )
26	24, 25	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ U = (/) ) -> X = U. (/) )
27		locfinref.x	. . . . . . 7 |- X = U. J
28	26, 27, 19	3eqtr3g 2679	. . . . . 6 |- ( ( ph /\ U = (/) ) -> U. J = (/) )
29		locfinref.5	. . . . . . . 8 |- ( ph -> V e. ( LocFin ` J ) )
30		locfintop 21324	. . . . . . . 8 |- ( V e. ( LocFin ` J ) -> J e. Top )
31		0top 20787	. . . . . . . 8 |- ( J e. Top -> ( U. J = (/) <-> J = { (/) } ) )
32	29, 30, 31	3syl 18	. . . . . . 7 |- ( ph -> ( U. J = (/) <-> J = { (/) } ) )
33	32	adantr 481	. . . . . 6 |- ( ( ph /\ U = (/) ) -> ( U. J = (/) <-> J = { (/) } ) )
34	28, 33	mpbid 222	. . . . 5 |- ( ( ph /\ U = (/) ) -> J = { (/) } )
35	34	fveq2d 6195	. . . 4 |- ( ( ph /\ U = (/) ) -> ( LocFin ` J ) = ( LocFin ` { (/) } ) )
36	22, 35	eleqtrrd 2704	. . 3 |- ( ( ph /\ U = (/) ) -> ran (/) e. ( LocFin ` J ) )
37		feq1 6026	. . . . 5 |- ( f = (/) -> ( f : U --> J <-> (/) : U --> J ) )
38		rneq 5351	. . . . . 6 |- ( f = (/) -> ran f = ran (/) )
39	38	breq1d 4663	. . . . 5 |- ( f = (/) -> ( ran f Ref U <-> ran (/) Ref U ) )
40	38	eleq1d 2686	. . . . 5 |- ( f = (/) -> ( ran f e. ( LocFin ` J ) <-> ran (/) e. ( LocFin ` J ) ) )
41	37, 39, 40	3anbi123d 1399	. . . 4 |- ( f = (/) -> ( ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) <-> ( (/) : U --> J /\ ran (/) Ref U /\ ran (/) e. ( LocFin ` J ) ) ) )
42	6, 41	spcev 3300	. . 3 |- ( ( (/) : U --> J /\ ran (/) Ref U /\ ran (/) e. ( LocFin ` J ) ) -> E. f ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) )
43	4, 10, 36, 42	syl3anc 1326	. 2 |- ( ( ph /\ U = (/) ) -> E. f ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) )
44		locfinref.1	. . . . 5 |- ( ph -> U C_ J )
45		locfinref.3	. . . . 5 |- ( ph -> V C_ J )
46		locfinref.4	. . . . 5 |- ( ph -> V Ref U )
47	27, 44, 23, 45, 46, 29	locfinreflem 29907	. . . 4 |- ( ph -> E. g ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) )
48	47	adantr 481	. . 3 |- ( ( ph /\ U =/= (/) ) -> E. g ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) )
49		simpl 473	. . . 4 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( ph /\ U =/= (/) ) )
50		simprl1 1106	. . . . . . . 8 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> Fun g )
51		fdmrn 6064	. . . . . . . 8 |- ( Fun g <-> g : dom g --> ran g )
52	50, 51	sylib 208	. . . . . . 7 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> g : dom g --> ran g )
53		simprl3 1108	. . . . . . 7 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ran g C_ J )
54	52, 53	fssd 6057	. . . . . 6 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> g : dom g --> J )
55		fconstg 6092	. . . . . . . 8 |- ( (/) e. _V -> ( ( U \ dom g ) X. { (/) } ) : ( U \ dom g ) --> { (/) } )
56	6, 55	mp1i 13	. . . . . . 7 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( ( U \ dom g ) X. { (/) } ) : ( U \ dom g ) --> { (/) } )
57		0opn 20709	. . . . . . . . . 10 |- ( J e. Top -> (/) e. J )
58	29, 30, 57	3syl 18	. . . . . . . . 9 |- ( ph -> (/) e. J )
59	58	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> (/) e. J )
60	59	snssd 4340	. . . . . . 7 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> { (/) } C_ J )
61	56, 60	fssd 6057	. . . . . 6 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( ( U \ dom g ) X. { (/) } ) : ( U \ dom g ) --> J )
62		disjdif 4040	. . . . . . 7 |- ( dom g i^i ( U \ dom g ) ) = (/)
63	62	a1i 11	. . . . . 6 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( dom g i^i ( U \ dom g ) ) = (/) )
64		fun2 6067	. . . . . 6 |- ( ( ( g : dom g --> J /\ ( ( U \ dom g ) X. { (/) } ) : ( U \ dom g ) --> J ) /\ ( dom g i^i ( U \ dom g ) ) = (/) ) -> ( g u. ( ( U \ dom g ) X. { (/) } ) ) : ( dom g u. ( U \ dom g ) ) --> J )
65	54, 61, 63, 64	syl21anc 1325	. . . . 5 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( g u. ( ( U \ dom g ) X. { (/) } ) ) : ( dom g u. ( U \ dom g ) ) --> J )
66		simprl2 1107	. . . . . . 7 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> dom g C_ U )
67		undif 4049	. . . . . . 7 |- ( dom g C_ U <-> ( dom g u. ( U \ dom g ) ) = U )
68	66, 67	sylib 208	. . . . . 6 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( dom g u. ( U \ dom g ) ) = U )
69	68	feq2d 6031	. . . . 5 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( ( g u. ( ( U \ dom g ) X. { (/) } ) ) : ( dom g u. ( U \ dom g ) ) --> J <-> ( g u. ( ( U \ dom g ) X. { (/) } ) ) : U --> J ) )
70	65, 69	mpbid 222	. . . 4 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( g u. ( ( U \ dom g ) X. { (/) } ) ) : U --> J )
71		simpr 477	. . . . . 6 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ran g ) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ran g )
72		simprrl 804	. . . . . . 7 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ran g Ref U )
73	72	adantr 481	. . . . . 6 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ran g ) -> ran g Ref U )
74	71, 73	eqbrtrd 4675	. . . . 5 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ran g ) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) Ref U )
75		simpr 477	. . . . . 6 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) )
76	49	simprd 479	. . . . . . . 8 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> U =/= (/) )
77		refun0 21318	. . . . . . . 8 |- ( ( ran g Ref U /\ U =/= (/) ) -> ( ran g u. { (/) } ) Ref U )
78	72, 76, 77	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( ran g u. { (/) } ) Ref U )
79	78	adantr 481	. . . . . 6 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> ( ran g u. { (/) } ) Ref U )
80	75, 79	eqbrtrd 4675	. . . . 5 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) Ref U )
81		rnxpss 5566	. . . . . . 7 |- ran ( ( U \ dom g ) X. { (/) } ) C_ { (/) }
82		sssn 4358	. . . . . . 7 |- ( ran ( ( U \ dom g ) X. { (/) } ) C_ { (/) } <-> ( ran ( ( U \ dom g ) X. { (/) } ) = (/) \/ ran ( ( U \ dom g ) X. { (/) } ) = { (/) } ) )
83	81, 82	mpbi 220	. . . . . 6 |- ( ran ( ( U \ dom g ) X. { (/) } ) = (/) \/ ran ( ( U \ dom g ) X. { (/) } ) = { (/) } )
84		rnun 5541	. . . . . . . . 9 |- ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. ran ( ( U \ dom g ) X. { (/) } ) )
85		uneq2 3761	. . . . . . . . 9 |- ( ran ( ( U \ dom g ) X. { (/) } ) = (/) -> ( ran g u. ran ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. (/) ) )
86	84, 85	syl5eq 2668	. . . . . . . 8 |- ( ran ( ( U \ dom g ) X. { (/) } ) = (/) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. (/) ) )
87		un0 3967	. . . . . . . 8 |- ( ran g u. (/) ) = ran g
88	86, 87	syl6eq 2672	. . . . . . 7 |- ( ran ( ( U \ dom g ) X. { (/) } ) = (/) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ran g )
89		uneq2 3761	. . . . . . . 8 |- ( ran ( ( U \ dom g ) X. { (/) } ) = { (/) } -> ( ran g u. ran ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) )
90	84, 89	syl5eq 2668	. . . . . . 7 |- ( ran ( ( U \ dom g ) X. { (/) } ) = { (/) } -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) )
91	88, 90	orim12i 538	. . . . . 6 |- ( ( ran ( ( U \ dom g ) X. { (/) } ) = (/) \/ ran ( ( U \ dom g ) X. { (/) } ) = { (/) } ) -> ( ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ran g \/ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) )
92	83, 91	mp1i 13	. . . . 5 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ( ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ran g \/ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) )
93	74, 80, 92	mpjaodan 827	. . . 4 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) Ref U )
94		simprrr 805	. . . . . . 7 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ran g e. ( LocFin ` J ) )
95	94	adantr 481	. . . . . 6 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ran g ) -> ran g e. ( LocFin ` J ) )
96	71, 95	eqeltrd 2701	. . . . 5 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ran g ) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. ( LocFin ` J ) )
97	94	adantr 481	. . . . . . 7 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> ran g e. ( LocFin ` J ) )
98		snfi 8038	. . . . . . . 8 |- { (/) } e. Fin
99	98	a1i 11	. . . . . . 7 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> { (/) } e. Fin )
100	59	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> (/) e. J )
101	100	snssd 4340	. . . . . . . 8 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> { (/) } C_ J )
102	101	unissd 4462	. . . . . . 7 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> U. { (/) } C_ U. J )
103		lfinun 21328	. . . . . . 7 |- ( ( ran g e. ( LocFin ` J ) /\ { (/) } e. Fin /\ U. { (/) } C_ U. J ) -> ( ran g u. { (/) } ) e. ( LocFin ` J ) )
104	97, 99, 102, 103	syl3anc 1326	. . . . . 6 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> ( ran g u. { (/) } ) e. ( LocFin ` J ) )
105	75, 104	eqeltrd 2701	. . . . 5 |- ( ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) = ( ran g u. { (/) } ) ) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. ( LocFin ` J ) )
106	96, 105, 92	mpjaodan 827	. . . 4 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. ( LocFin ` J ) )
107		refrel 21311	. . . . . . . . 9 |- Rel Ref
108	107	brrelex2i 5159	. . . . . . . 8 |- ( V Ref U -> U e. _V )
109		difexg 4808	. . . . . . . 8 |- ( U e. _V -> ( U \ dom g ) e. _V )
110	46, 108, 109	3syl 18	. . . . . . 7 |- ( ph -> ( U \ dom g ) e. _V )
111	110	adantr 481	. . . . . 6 |- ( ( ph /\ U =/= (/) ) -> ( U \ dom g ) e. _V )
112		p0ex 4853	. . . . . . 7 |- { (/) } e. _V
113		xpexg 6960	. . . . . . 7 |- ( ( ( U \ dom g ) e. _V /\ { (/) } e. _V ) -> ( ( U \ dom g ) X. { (/) } ) e. _V )
114	112, 113	mpan2 707	. . . . . 6 |- ( ( U \ dom g ) e. _V -> ( ( U \ dom g ) X. { (/) } ) e. _V )
115		vex 3203	. . . . . . 7 |- g e. _V
116		unexg 6959	. . . . . . 7 |- ( ( g e. _V /\ ( ( U \ dom g ) X. { (/) } ) e. _V ) -> ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. _V )
117	115, 116	mpan 706	. . . . . 6 |- ( ( ( U \ dom g ) X. { (/) } ) e. _V -> ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. _V )
118		feq1 6026	. . . . . . . 8 |- ( f = ( g u. ( ( U \ dom g ) X. { (/) } ) ) -> ( f : U --> J <-> ( g u. ( ( U \ dom g ) X. { (/) } ) ) : U --> J ) )
119		rneq 5351	. . . . . . . . 9 |- ( f = ( g u. ( ( U \ dom g ) X. { (/) } ) ) -> ran f = ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) )
120	119	breq1d 4663	. . . . . . . 8 |- ( f = ( g u. ( ( U \ dom g ) X. { (/) } ) ) -> ( ran f Ref U <-> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) Ref U ) )
121	119	eleq1d 2686	. . . . . . . 8 |- ( f = ( g u. ( ( U \ dom g ) X. { (/) } ) ) -> ( ran f e. ( LocFin ` J ) <-> ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. ( LocFin ` J ) ) )
122	118, 120, 121	3anbi123d 1399	. . . . . . 7 |- ( f = ( g u. ( ( U \ dom g ) X. { (/) } ) ) -> ( ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) <-> ( ( g u. ( ( U \ dom g ) X. { (/) } ) ) : U --> J /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) Ref U /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. ( LocFin ` J ) ) ) )
123	122	spcegv 3294	. . . . . 6 |- ( ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. _V -> ( ( ( g u. ( ( U \ dom g ) X. { (/) } ) ) : U --> J /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) Ref U /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. ( LocFin ` J ) ) -> E. f ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) ) )
124	111, 114, 117, 123	4syl 19	. . . . 5 |- ( ( ph /\ U =/= (/) ) -> ( ( ( g u. ( ( U \ dom g ) X. { (/) } ) ) : U --> J /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) Ref U /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. ( LocFin ` J ) ) -> E. f ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) ) )
125	124	imp 445	. . . 4 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( g u. ( ( U \ dom g ) X. { (/) } ) ) : U --> J /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) Ref U /\ ran ( g u. ( ( U \ dom g ) X. { (/) } ) ) e. ( LocFin ` J ) ) ) -> E. f ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) )
126	49, 70, 93, 106, 125	syl13anc 1328	. . 3 |- ( ( ( ph /\ U =/= (/) ) /\ ( ( Fun g /\ dom g C_ U /\ ran g C_ J ) /\ ( ran g Ref U /\ ran g e. ( LocFin ` J ) ) ) ) -> E. f ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) )
127	48, 126	exlimddv 1863	. 2 |- ( ( ph /\ U =/= (/) ) -> E. f ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) )
128	43, 127	pm2.61dane 2881	1 |- ( ph -> E. f ( f : U --> J /\ ran f Ref U /\ ran f e. ( LocFin ` J ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   Fun wfun 5882   -->wf 5884   ` cfv 5888   Fincfn 7955   Topctop 20698   Refcref 21305   LocFinclocfin 21307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-r1 8627  df-rank 8628  df-card 8765  df-ac 8939  df-top 20699  df-topon 20716  df-ref 21308  df-locfin 21310

This theorem is referenced by:  pcmplfinf  29928