Theorem bndth 22757

Description: The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to -u F.) (Contributed by Mario Carneiro, 12-Aug-2014.)

Hypotheses

Ref	Expression
bndth.1	|- X = U. J
bndth.2	|- K = ( topGen ` ran (,) )
bndth.3	|- ( ph -> J e. Comp )
bndth.4	|- ( ph -> F e. ( J Cn K ) )

Assertion

Ref	Expression
bndth	|- ( ph -> E. x e. RR A. y e. X ( F ` y ) <_ x )

Distinct variable groups:   x, y, F   y, K   ph, x, y   x, X, y   x, J, y

Allowed substitution hint:   K( x)

Proof of Theorem bndth

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

Step	Hyp	Ref	Expression
1		bndth.4	. . . . 5 |- ( ph -> F e. ( J Cn K ) )
2		bndth.1	. . . . . 6 |- X = U. J
3		bndth.2	. . . . . . . 8 |- K = ( topGen ` ran (,) )
4		retopon 22567	. . . . . . . 8 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
5	3, 4	eqeltri 2697	. . . . . . 7 |- K e. (TopOn ` RR )
6	5	toponunii 20721	. . . . . 6 |- RR = U. K
7	2, 6	cnf 21050	. . . . 5 |- ( F e. ( J Cn K ) -> F : X --> RR )
8	1, 7	syl 17	. . . 4 |- ( ph -> F : X --> RR )
9		frn 6053	. . . 4 |- ( F : X --> RR -> ran F C_ RR )
10	8, 9	syl 17	. . 3 |- ( ph -> ran F C_ RR )
11		imassrn 5477	. . . . . 6 |- ( (,) " ( { -oo } X. RR ) ) C_ ran (,)
12		retopbas 22564	. . . . . . . 8 |- ran (,) e. TopBases
13		bastg 20770	. . . . . . . 8 |- ( ran (,) e. TopBases -> ran (,) C_ ( topGen ` ran (,) ) )
14	12, 13	ax-mp 5	. . . . . . 7 |- ran (,) C_ ( topGen ` ran (,) )
15	14, 3	sseqtr4i 3638	. . . . . 6 |- ran (,) C_ K
16	11, 15	sstri 3612	. . . . 5 |- ( (,) " ( { -oo } X. RR ) ) C_ K
17		retop 22565	. . . . . . . 8 |- ( topGen ` ran (,) ) e. Top
18	3, 17	eqeltri 2697	. . . . . . 7 |- K e. Top
19	18	elexi 3213	. . . . . 6 |- K e. _V
20	19	elpw2 4828	. . . . 5 |- ( ( (,) " ( { -oo } X. RR ) ) e. ~P K <-> ( (,) " ( { -oo } X. RR ) ) C_ K )
21	16, 20	mpbir 221	. . . 4 |- ( (,) " ( { -oo } X. RR ) ) e. ~P K
22		bndth.3	. . . . . 6 |- ( ph -> J e. Comp )
23		rncmp 21199	. . . . . 6 |- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ( K ↾t ran F ) e. Comp )
24	22, 1, 23	syl2anc 693	. . . . 5 |- ( ph -> ( K ↾t ran F ) e. Comp )
25	6	cmpsub 21203	. . . . . 6 |- ( ( K e. Top /\ ran F C_ RR ) -> ( ( K ↾t ran F ) e. Comp <-> A. u e. ~P K ( ran F C_ U. u -> E. v e. ( ~P u i^i Fin ) ran F C_ U. v ) ) )
26	18, 10, 25	sylancr 695	. . . . 5 |- ( ph -> ( ( K ↾t ran F ) e. Comp <-> A. u e. ~P K ( ran F C_ U. u -> E. v e. ( ~P u i^i Fin ) ran F C_ U. v ) ) )
27	24, 26	mpbid 222	. . . 4 |- ( ph -> A. u e. ~P K ( ran F C_ U. u -> E. v e. ( ~P u i^i Fin ) ran F C_ U. v ) )
28		unieq 4444	. . . . . . . 8 |- ( u = ( (,) " ( { -oo } X. RR ) ) -> U. u = U. ( (,) " ( { -oo } X. RR ) ) )
29	11	unissi 4461	. . . . . . . . . 10 |- U. ( (,) " ( { -oo } X. RR ) ) C_ U. ran (,)
30		unirnioo 12273	. . . . . . . . . 10 |- RR = U. ran (,)
31	29, 30	sseqtr4i 3638	. . . . . . . . 9 |- U. ( (,) " ( { -oo } X. RR ) ) C_ RR
32		id 22	. . . . . . . . . . . 12 |- ( x e. RR -> x e. RR )
33		ltp1 10861	. . . . . . . . . . . 12 |- ( x e. RR -> x < ( x + 1 ) )
34		ressxr 10083	. . . . . . . . . . . . . 14 |- RR C_ RR*
35		peano2re 10209	. . . . . . . . . . . . . 14 |- ( x e. RR -> ( x + 1 ) e. RR )
36	34, 35	sseldi 3601	. . . . . . . . . . . . 13 |- ( x e. RR -> ( x + 1 ) e. RR* )
37		elioomnf 12268	. . . . . . . . . . . . 13 |- ( ( x + 1 ) e. RR* -> ( x e. ( -oo (,) ( x + 1 ) ) <-> ( x e. RR /\ x < ( x + 1 ) ) ) )
38	36, 37	syl 17	. . . . . . . . . . . 12 |- ( x e. RR -> ( x e. ( -oo (,) ( x + 1 ) ) <-> ( x e. RR /\ x < ( x + 1 ) ) ) )
39	32, 33, 38	mpbir2and 957	. . . . . . . . . . 11 |- ( x e. RR -> x e. ( -oo (,) ( x + 1 ) ) )
40		df-ov 6653	. . . . . . . . . . . 12 |- ( -oo (,) ( x + 1 ) ) = ( (,) ` <. -oo , ( x + 1 ) >. )
41		mnfxr 10096	. . . . . . . . . . . . . . . 16 |- -oo e. RR*
42	41	elexi 3213	. . . . . . . . . . . . . . 15 |- -oo e. _V
43	42	snid 4208	. . . . . . . . . . . . . 14 |- -oo e. { -oo }
44		opelxpi 5148	. . . . . . . . . . . . . 14 |- ( ( -oo e. { -oo } /\ ( x + 1 ) e. RR ) -> <. -oo , ( x + 1 ) >. e. ( { -oo } X. RR ) )
45	43, 35, 44	sylancr 695	. . . . . . . . . . . . 13 |- ( x e. RR -> <. -oo , ( x + 1 ) >. e. ( { -oo } X. RR ) )
46		ioof 12271	. . . . . . . . . . . . . . 15 |- (,) : ( RR* X. RR* ) --> ~P RR
47		ffun 6048	. . . . . . . . . . . . . . 15 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> Fun (,) )
48	46, 47	ax-mp 5	. . . . . . . . . . . . . 14 |- Fun (,)
49		snssi 4339	. . . . . . . . . . . . . . . . 17 |- ( -oo e. RR* -> { -oo } C_ RR* )
50	41, 49	ax-mp 5	. . . . . . . . . . . . . . . 16 |- { -oo } C_ RR*
51		xpss12 5225	. . . . . . . . . . . . . . . 16 |- ( ( { -oo } C_ RR* /\ RR C_ RR* ) -> ( { -oo } X. RR ) C_ ( RR* X. RR* ) )
52	50, 34, 51	mp2an 708	. . . . . . . . . . . . . . 15 |- ( { -oo } X. RR ) C_ ( RR* X. RR* )
53	46	fdmi 6052	. . . . . . . . . . . . . . 15 |- dom (,) = ( RR* X. RR* )
54	52, 53	sseqtr4i 3638	. . . . . . . . . . . . . 14 |- ( { -oo } X. RR ) C_ dom (,)
55		funfvima2 6493	. . . . . . . . . . . . . 14 |- ( ( Fun (,) /\ ( { -oo } X. RR ) C_ dom (,) ) -> ( <. -oo , ( x + 1 ) >. e. ( { -oo } X. RR ) -> ( (,) ` <. -oo , ( x + 1 ) >. ) e. ( (,) " ( { -oo } X. RR ) ) ) )
56	48, 54, 55	mp2an 708	. . . . . . . . . . . . 13 |- ( <. -oo , ( x + 1 ) >. e. ( { -oo } X. RR ) -> ( (,) ` <. -oo , ( x + 1 ) >. ) e. ( (,) " ( { -oo } X. RR ) ) )
57	45, 56	syl 17	. . . . . . . . . . . 12 |- ( x e. RR -> ( (,) ` <. -oo , ( x + 1 ) >. ) e. ( (,) " ( { -oo } X. RR ) ) )
58	40, 57	syl5eqel 2705	. . . . . . . . . . 11 |- ( x e. RR -> ( -oo (,) ( x + 1 ) ) e. ( (,) " ( { -oo } X. RR ) ) )
59		elunii 4441	. . . . . . . . . . 11 |- ( ( x e. ( -oo (,) ( x + 1 ) ) /\ ( -oo (,) ( x + 1 ) ) e. ( (,) " ( { -oo } X. RR ) ) ) -> x e. U. ( (,) " ( { -oo } X. RR ) ) )
60	39, 58, 59	syl2anc 693	. . . . . . . . . 10 |- ( x e. RR -> x e. U. ( (,) " ( { -oo } X. RR ) ) )
61	60	ssriv 3607	. . . . . . . . 9 |- RR C_ U. ( (,) " ( { -oo } X. RR ) )
62	31, 61	eqssi 3619	. . . . . . . 8 |- U. ( (,) " ( { -oo } X. RR ) ) = RR
63	28, 62	syl6eq 2672	. . . . . . 7 |- ( u = ( (,) " ( { -oo } X. RR ) ) -> U. u = RR )
64	63	sseq2d 3633	. . . . . 6 |- ( u = ( (,) " ( { -oo } X. RR ) ) -> ( ran F C_ U. u <-> ran F C_ RR ) )
65		pweq 4161	. . . . . . . 8 |- ( u = ( (,) " ( { -oo } X. RR ) ) -> ~P u = ~P ( (,) " ( { -oo } X. RR ) ) )
66	65	ineq1d 3813	. . . . . . 7 |- ( u = ( (,) " ( { -oo } X. RR ) ) -> ( ~P u i^i Fin ) = ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) )
67	66	rexeqdv 3145	. . . . . 6 |- ( u = ( (,) " ( { -oo } X. RR ) ) -> ( E. v e. ( ~P u i^i Fin ) ran F C_ U. v <-> E. v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ran F C_ U. v ) )
68	64, 67	imbi12d 334	. . . . 5 |- ( u = ( (,) " ( { -oo } X. RR ) ) -> ( ( ran F C_ U. u -> E. v e. ( ~P u i^i Fin ) ran F C_ U. v ) <-> ( ran F C_ RR -> E. v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ran F C_ U. v ) ) )
69	68	rspcv 3305	. . . 4 |- ( ( (,) " ( { -oo } X. RR ) ) e. ~P K -> ( A. u e. ~P K ( ran F C_ U. u -> E. v e. ( ~P u i^i Fin ) ran F C_ U. v ) -> ( ran F C_ RR -> E. v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ran F C_ U. v ) ) )
70	21, 27, 69	mpsyl 68	. . 3 |- ( ph -> ( ran F C_ RR -> E. v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ran F C_ U. v ) )
71	10, 70	mpd 15	. 2 |- ( ph -> E. v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ran F C_ U. v )
72		simpr 477	. . . . . . 7 |- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) )
73		elin 3796	. . . . . . 7 |- ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) <-> ( v e. ~P ( (,) " ( { -oo } X. RR ) ) /\ v e. Fin ) )
74	72, 73	sylib 208	. . . . . 6 |- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> ( v e. ~P ( (,) " ( { -oo } X. RR ) ) /\ v e. Fin ) )
75	74	adantrr 753	. . . . 5 |- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> ( v e. ~P ( (,) " ( { -oo } X. RR ) ) /\ v e. Fin ) )
76	75	simprd 479	. . . 4 |- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> v e. Fin )
77	74	simpld 475	. . . . . . 7 |- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> v e. ~P ( (,) " ( { -oo } X. RR ) ) )
78	77	elpwid 4170	. . . . . 6 |- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> v C_ ( (,) " ( { -oo } X. RR ) ) )
79	50	sseli 3599	. . . . . . . . . . . 12 |- ( u e. { -oo } -> u e. RR* )
80	79	adantr 481	. . . . . . . . . . 11 |- ( ( u e. { -oo } /\ w e. RR ) -> u e. RR* )
81	34	sseli 3599	. . . . . . . . . . . 12 |- ( w e. RR -> w e. RR* )
82	81	adantl 482	. . . . . . . . . . 11 |- ( ( u e. { -oo } /\ w e. RR ) -> w e. RR* )
83		mnflt 11957	. . . . . . . . . . . . . . 15 |- ( w e. RR -> -oo < w )
84		xrltnle 10105	. . . . . . . . . . . . . . . 16 |- ( ( -oo e. RR* /\ w e. RR* ) -> ( -oo < w <-> -. w <_ -oo ) )
85	41, 81, 84	sylancr 695	. . . . . . . . . . . . . . 15 |- ( w e. RR -> ( -oo < w <-> -. w <_ -oo ) )
86	83, 85	mpbid 222	. . . . . . . . . . . . . 14 |- ( w e. RR -> -. w <_ -oo )
87	86	adantl 482	. . . . . . . . . . . . 13 |- ( ( u e. { -oo } /\ w e. RR ) -> -. w <_ -oo )
88		elsni 4194	. . . . . . . . . . . . . . 15 |- ( u e. { -oo } -> u = -oo )
89	88	adantr 481	. . . . . . . . . . . . . 14 |- ( ( u e. { -oo } /\ w e. RR ) -> u = -oo )
90	89	breq2d 4665	. . . . . . . . . . . . 13 |- ( ( u e. { -oo } /\ w e. RR ) -> ( w <_ u <-> w <_ -oo ) )
91	87, 90	mtbird 315	. . . . . . . . . . . 12 |- ( ( u e. { -oo } /\ w e. RR ) -> -. w <_ u )
92		ioo0 12200	. . . . . . . . . . . . . 14 |- ( ( u e. RR* /\ w e. RR* ) -> ( ( u (,) w ) = (/) <-> w <_ u ) )
93	79, 81, 92	syl2an 494	. . . . . . . . . . . . 13 |- ( ( u e. { -oo } /\ w e. RR ) -> ( ( u (,) w ) = (/) <-> w <_ u ) )
94	93	necon3abid 2830	. . . . . . . . . . . 12 |- ( ( u e. { -oo } /\ w e. RR ) -> ( ( u (,) w ) =/= (/) <-> -. w <_ u ) )
95	91, 94	mpbird 247	. . . . . . . . . . 11 |- ( ( u e. { -oo } /\ w e. RR ) -> ( u (,) w ) =/= (/) )
96		df-ioo 12179	. . . . . . . . . . . 12 |- (,) = ( y e. RR* , z e. RR* |-> { v e. RR* | ( y < v /\ v < z ) } )
97		idd 24	. . . . . . . . . . . 12 |- ( ( x e. RR* /\ w e. RR* ) -> ( x < w -> x < w ) )
98		xrltle 11982	. . . . . . . . . . . 12 |- ( ( x e. RR* /\ w e. RR* ) -> ( x < w -> x <_ w ) )
99		idd 24	. . . . . . . . . . . 12 |- ( ( u e. RR* /\ x e. RR* ) -> ( u < x -> u < x ) )
100		xrltle 11982	. . . . . . . . . . . 12 |- ( ( u e. RR* /\ x e. RR* ) -> ( u < x -> u <_ x ) )
101	96, 97, 98, 99, 100	ixxub 12196	. . . . . . . . . . 11 |- ( ( u e. RR* /\ w e. RR* /\ ( u (,) w ) =/= (/) ) -> sup ( ( u (,) w ) , RR* , < ) = w )
102	80, 82, 95, 101	syl3anc 1326	. . . . . . . . . 10 |- ( ( u e. { -oo } /\ w e. RR ) -> sup ( ( u (,) w ) , RR* , < ) = w )
103		simpr 477	. . . . . . . . . 10 |- ( ( u e. { -oo } /\ w e. RR ) -> w e. RR )
104	102, 103	eqeltrd 2701	. . . . . . . . 9 |- ( ( u e. { -oo } /\ w e. RR ) -> sup ( ( u (,) w ) , RR* , < ) e. RR )
105	104	rgen2 2975	. . . . . . . 8 |- A. u e. { -oo } A. w e. RR sup ( ( u (,) w ) , RR* , < ) e. RR
106		fveq2 6191	. . . . . . . . . . . 12 |- ( z = <. u , w >. -> ( (,) ` z ) = ( (,) ` <. u , w >. ) )
107		df-ov 6653	. . . . . . . . . . . 12 |- ( u (,) w ) = ( (,) ` <. u , w >. )
108	106, 107	syl6eqr 2674	. . . . . . . . . . 11 |- ( z = <. u , w >. -> ( (,) ` z ) = ( u (,) w ) )
109	108	supeq1d 8352	. . . . . . . . . 10 |- ( z = <. u , w >. -> sup ( ( (,) ` z ) , RR* , < ) = sup ( ( u (,) w ) , RR* , < ) )
110	109	eleq1d 2686	. . . . . . . . 9 |- ( z = <. u , w >. -> ( sup ( ( (,) ` z ) , RR* , < ) e. RR <-> sup ( ( u (,) w ) , RR* , < ) e. RR ) )
111	110	ralxp 5263	. . . . . . . 8 |- ( A. z e. ( { -oo } X. RR ) sup ( ( (,) ` z ) , RR* , < ) e. RR <-> A. u e. { -oo } A. w e. RR sup ( ( u (,) w ) , RR* , < ) e. RR )
112	105, 111	mpbir 221	. . . . . . 7 |- A. z e. ( { -oo } X. RR ) sup ( ( (,) ` z ) , RR* , < ) e. RR
113		ffn 6045	. . . . . . . . 9 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
114	46, 113	ax-mp 5	. . . . . . . 8 |- (,) Fn ( RR* X. RR* )
115		supeq1 8351	. . . . . . . . . 10 |- ( w = ( (,) ` z ) -> sup ( w , RR* , < ) = sup ( ( (,) ` z ) , RR* , < ) )
116	115	eleq1d 2686	. . . . . . . . 9 |- ( w = ( (,) ` z ) -> ( sup ( w , RR* , < ) e. RR <-> sup ( ( (,) ` z ) , RR* , < ) e. RR ) )
117	116	ralima 6498	. . . . . . . 8 |- ( ( (,) Fn ( RR* X. RR* ) /\ ( { -oo } X. RR ) C_ ( RR* X. RR* ) ) -> ( A. w e. ( (,) " ( { -oo } X. RR ) ) sup ( w , RR* , < ) e. RR <-> A. z e. ( { -oo } X. RR ) sup ( ( (,) ` z ) , RR* , < ) e. RR ) )
118	114, 52, 117	mp2an 708	. . . . . . 7 |- ( A. w e. ( (,) " ( { -oo } X. RR ) ) sup ( w , RR* , < ) e. RR <-> A. z e. ( { -oo } X. RR ) sup ( ( (,) ` z ) , RR* , < ) e. RR )
119	112, 118	mpbir 221	. . . . . 6 |- A. w e. ( (,) " ( { -oo } X. RR ) ) sup ( w , RR* , < ) e. RR
120		ssralv 3666	. . . . . 6 |- ( v C_ ( (,) " ( { -oo } X. RR ) ) -> ( A. w e. ( (,) " ( { -oo } X. RR ) ) sup ( w , RR* , < ) e. RR -> A. w e. v sup ( w , RR* , < ) e. RR ) )
121	78, 119, 120	mpisyl 21	. . . . 5 |- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> A. w e. v sup ( w , RR* , < ) e. RR )
122	121	adantrr 753	. . . 4 |- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> A. w e. v sup ( w , RR* , < ) e. RR )
123		fimaxre3 10970	. . . 4 |- ( ( v e. Fin /\ A. w e. v sup ( w , RR* , < ) e. RR ) -> E. x e. RR A. w e. v sup ( w , RR* , < ) <_ x )
124	76, 122, 123	syl2anc 693	. . 3 |- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> E. x e. RR A. w e. v sup ( w , RR* , < ) <_ x )
125		simplrr 801	. . . . . . . 8 |- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ran F C_ U. v )
126	125	sselda 3603	. . . . . . 7 |- ( ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) /\ z e. ran F ) -> z e. U. v )
127		eluni2 4440	. . . . . . . 8 |- ( z e. U. v <-> E. w e. v z e. w )
128		r19.29r 3073	. . . . . . . . . 10 |- ( ( E. w e. v z e. w /\ A. w e. v sup ( w , RR* , < ) <_ x ) -> E. w e. v ( z e. w /\ sup ( w , RR* , < ) <_ x ) )
129		sspwuni 4611	. . . . . . . . . . . . . . . . . . 19 |- ( ( (,) " ( { -oo } X. RR ) ) C_ ~P RR <-> U. ( (,) " ( { -oo } X. RR ) ) C_ RR )
130	31, 129	mpbir 221	. . . . . . . . . . . . . . . . . 18 |- ( (,) " ( { -oo } X. RR ) ) C_ ~P RR
131	78	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> v C_ ( (,) " ( { -oo } X. RR ) ) )
132		simp2r 1088	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w e. v )
133	131, 132	sseldd 3604	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w e. ( (,) " ( { -oo } X. RR ) ) )
134	130, 133	sseldi 3601	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w e. ~P RR )
135	134	elpwid 4170	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w C_ RR )
136		simp3l 1089	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> z e. w )
137	135, 136	sseldd 3604	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> z e. RR )
138	121	r19.21bi 2932	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ w e. v ) -> sup ( w , RR* , < ) e. RR )
139	138	adantrl 752	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) ) -> sup ( w , RR* , < ) e. RR )
140	139	3adant3 1081	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> sup ( w , RR* , < ) e. RR )
141		simp2l 1087	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> x e. RR )
142	135, 34	syl6ss 3615	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w C_ RR* )
143		supxrub 12154	. . . . . . . . . . . . . . . 16 |- ( ( w C_ RR* /\ z e. w ) -> z <_ sup ( w , RR* , < ) )
144	142, 136, 143	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> z <_ sup ( w , RR* , < ) )
145		simp3r 1090	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> sup ( w , RR* , < ) <_ x )
146	137, 140, 141, 144, 145	letrd 10194	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> z <_ x )
147	146	3expia 1267	. . . . . . . . . . . . 13 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) ) -> ( ( z e. w /\ sup ( w , RR* , < ) <_ x ) -> z <_ x ) )
148	147	anassrs 680	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ x e. RR ) /\ w e. v ) -> ( ( z e. w /\ sup ( w , RR* , < ) <_ x ) -> z <_ x ) )
149	148	rexlimdva 3031	. . . . . . . . . . 11 |- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ x e. RR ) -> ( E. w e. v ( z e. w /\ sup ( w , RR* , < ) <_ x ) -> z <_ x ) )
150	149	adantlrr 757	. . . . . . . . . 10 |- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( E. w e. v ( z e. w /\ sup ( w , RR* , < ) <_ x ) -> z <_ x ) )
151	128, 150	syl5 34	. . . . . . . . 9 |- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( ( E. w e. v z e. w /\ A. w e. v sup ( w , RR* , < ) <_ x ) -> z <_ x ) )
152	151	expdimp 453	. . . . . . . 8 |- ( ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) /\ E. w e. v z e. w ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> z <_ x ) )
153	127, 152	sylan2b 492	. . . . . . 7 |- ( ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) /\ z e. U. v ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> z <_ x ) )
154	126, 153	syldan 487	. . . . . 6 |- ( ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) /\ z e. ran F ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> z <_ x ) )
155	154	ralrimdva 2969	. . . . 5 |- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> A. z e. ran F z <_ x ) )
156		ffn 6045	. . . . . . . 8 |- ( F : X --> RR -> F Fn X )
157	8, 156	syl 17	. . . . . . 7 |- ( ph -> F Fn X )
158	157	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> F Fn X )
159		breq1 4656	. . . . . . 7 |- ( z = ( F ` y ) -> ( z <_ x <-> ( F ` y ) <_ x ) )
160	159	ralrn 6362	. . . . . 6 |- ( F Fn X -> ( A. z e. ran F z <_ x <-> A. y e. X ( F ` y ) <_ x ) )
161	158, 160	syl 17	. . . . 5 |- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( A. z e. ran F z <_ x <-> A. y e. X ( F ` y ) <_ x ) )
162	155, 161	sylibd 229	. . . 4 |- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> A. y e. X ( F ` y ) <_ x ) )
163	162	reximdva 3017	. . 3 |- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> ( E. x e. RR A. w e. v sup ( w , RR* , < ) <_ x -> E. x e. RR A. y e. X ( F ` y ) <_ x ) )
164	124, 163	mpd 15	. 2 |- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> E. x e. RR A. y e. X ( F ` y ) <_ x )
165	71, 164	rexlimddv 3035	1 |- ( ph -> E. x e. RR A. y e. X ( F ` y ) <_ x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   supcsup 8346   RRcr 9935   1c1 9937   + caddc 9939   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   ↾_t crest 16081   topGenctg 16098   Topctop 20698  TopOnctopon 20715   TopBasesctb 20749   Cn ccn 21028   Compccmp 21189

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-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-nel 2898  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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ioo 12179  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cmp 21190

This theorem is referenced by:  evth  22758