Theorem smfpimbor1lem1 41005

Description: Every open set belongs to T. This is the second step in the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smfpimbor1lem1.s	|- ( ph -> S e. SAlg )
smfpimbor1lem1.f	|- ( ph -> F e. (SMblFn ` S ) )
smfpimbor1lem1.a	|- D = dom F
smfpimbor1lem1.j	|- J = ( topGen ` ran (,) )
smfpimbor1lem1.8	|- ( ph -> G e. J )
smfpimbor1lem1.t	|- T = { e e. ~P RR | ( `' F " e ) e. ( S ↾t D ) }

Assertion

Ref	Expression
smfpimbor1lem1	|- ( ph -> G e. T )

Distinct variable groups:   D, e   e, F   S, e   ph, e

Allowed substitution hints:   T( e)   G( e)   J( e)

Proof of Theorem smfpimbor1lem1

Dummy variables q p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfpimbor1lem1.j	. . 3 |- J = ( topGen ` ran (,) )
2		smfpimbor1lem1.8	. . 3 |- ( ph -> G e. J )
3	1, 2	tgqioo2 39774	. 2 |- ( ph -> E. q ( q C_ ( (,) " ( QQ X. QQ ) ) /\ G = U. q ) )
4		simprr 796	. . . . 5 |- ( ( ph /\ ( q C_ ( (,) " ( QQ X. QQ ) ) /\ G = U. q ) ) -> G = U. q )
5		smfpimbor1lem1.s	. . . . . . . . 9 |- ( ph -> S e. SAlg )
6		smfpimbor1lem1.f	. . . . . . . . 9 |- ( ph -> F e. (SMblFn ` S ) )
7		smfpimbor1lem1.a	. . . . . . . . 9 |- D = dom F
8		smfpimbor1lem1.t	. . . . . . . . 9 |- T = { e e. ~P RR | ( `' F " e ) e. ( S ↾t D ) }
9	5, 6, 7, 8	smfresal 40995	. . . . . . . 8 |- ( ph -> T e. SAlg )
10	9	adantr 481	. . . . . . 7 |- ( ( ph /\ q C_ ( (,) " ( QQ X. QQ ) ) ) -> T e. SAlg )
11		iooex 12198	. . . . . . . . . . . 12 |- (,) e. _V
12	11	imaexi 39415	. . . . . . . . . . 11 |- ( (,) " ( QQ X. QQ ) ) e. _V
13	12	a1i 11	. . . . . . . . . 10 |- ( q C_ ( (,) " ( QQ X. QQ ) ) -> ( (,) " ( QQ X. QQ ) ) e. _V )
14		id 22	. . . . . . . . . 10 |- ( q C_ ( (,) " ( QQ X. QQ ) ) -> q C_ ( (,) " ( QQ X. QQ ) ) )
15	13, 14	ssexd 4805	. . . . . . . . 9 |- ( q C_ ( (,) " ( QQ X. QQ ) ) -> q e. _V )
16	15	adantl 482	. . . . . . . 8 |- ( ( ph /\ q C_ ( (,) " ( QQ X. QQ ) ) ) -> q e. _V )
17		simpr 477	. . . . . . . . 9 |- ( ( ph /\ q C_ ( (,) " ( QQ X. QQ ) ) ) -> q C_ ( (,) " ( QQ X. QQ ) ) )
18		ioofun 39778	. . . . . . . . . . . . . . 15 |- Fun (,)
19	18	a1i 11	. . . . . . . . . . . . . 14 |- ( q e. ( (,) " ( QQ X. QQ ) ) -> Fun (,) )
20		id 22	. . . . . . . . . . . . . 14 |- ( q e. ( (,) " ( QQ X. QQ ) ) -> q e. ( (,) " ( QQ X. QQ ) ) )
21		fvelima 6248	. . . . . . . . . . . . . 14 |- ( ( Fun (,) /\ q e. ( (,) " ( QQ X. QQ ) ) ) -> E. p e. ( QQ X. QQ ) ( (,) ` p ) = q )
22	19, 20, 21	syl2anc 693	. . . . . . . . . . . . 13 |- ( q e. ( (,) " ( QQ X. QQ ) ) -> E. p e. ( QQ X. QQ ) ( (,) ` p ) = q )
23	22	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ q e. ( (,) " ( QQ X. QQ ) ) ) -> E. p e. ( QQ X. QQ ) ( (,) ` p ) = q )
24		id 22	. . . . . . . . . . . . . . . . . . . 20 |- ( ( (,) ` p ) = q -> ( (,) ` p ) = q )
25	24	eqcomd 2628	. . . . . . . . . . . . . . . . . . 19 |- ( ( (,) ` p ) = q -> q = ( (,) ` p ) )
26	25	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( p e. ( QQ X. QQ ) /\ ( (,) ` p ) = q ) -> q = ( (,) ` p ) )
27		1st2nd2 7205	. . . . . . . . . . . . . . . . . . . . 21 |- ( p e. ( QQ X. QQ ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
28	27	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 |- ( p e. ( QQ X. QQ ) -> ( (,) ` p ) = ( (,) ` <. ( 1st ` p ) , ( 2nd ` p ) >. ) )
29		df-ov 6653	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 1st ` p ) (,) ( 2nd ` p ) ) = ( (,) ` <. ( 1st ` p ) , ( 2nd ` p ) >. )
30	29	eqcomi 2631	. . . . . . . . . . . . . . . . . . . . 21 |- ( (,) ` <. ( 1st ` p ) , ( 2nd ` p ) >. ) = ( ( 1st ` p ) (,) ( 2nd ` p ) )
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( p e. ( QQ X. QQ ) -> ( (,) ` <. ( 1st ` p ) , ( 2nd ` p ) >. ) = ( ( 1st ` p ) (,) ( 2nd ` p ) ) )
32	28, 31	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 |- ( p e. ( QQ X. QQ ) -> ( (,) ` p ) = ( ( 1st ` p ) (,) ( 2nd ` p ) ) )
33	32	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( p e. ( QQ X. QQ ) /\ ( (,) ` p ) = q ) -> ( (,) ` p ) = ( ( 1st ` p ) (,) ( 2nd ` p ) ) )
34	26, 33	eqtrd 2656	. . . . . . . . . . . . . . . . 17 |- ( ( p e. ( QQ X. QQ ) /\ ( (,) ` p ) = q ) -> q = ( ( 1st ` p ) (,) ( 2nd ` p ) ) )
35	34	3adant1 1079	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( QQ X. QQ ) /\ ( (,) ` p ) = q ) -> q = ( ( 1st ` p ) (,) ( 2nd ` p ) ) )
36		ioossre 12235	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 1st ` p ) (,) ( 2nd ` p ) ) C_ RR
37		ovex 6678	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 1st ` p ) (,) ( 2nd ` p ) ) e. _V
38	37	elpw 4164	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( 1st ` p ) (,) ( 2nd ` p ) ) e. ~P RR <-> ( ( 1st ` p ) (,) ( 2nd ` p ) ) C_ RR )
39	36, 38	mpbir 221	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 1st ` p ) (,) ( 2nd ` p ) ) e. ~P RR
40	39	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( QQ X. QQ ) ) -> ( ( 1st ` p ) (,) ( 2nd ` p ) ) e. ~P RR )
41	5	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( QQ X. QQ ) ) -> S e. SAlg )
42	6	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( QQ X. QQ ) ) -> F e. (SMblFn ` S ) )
43		xp1st 7198	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( p e. ( QQ X. QQ ) -> ( 1st ` p ) e. QQ )
44	43	qred 39606	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( QQ X. QQ ) -> ( 1st ` p ) e. RR )
45	44	rexrd 10089	. . . . . . . . . . . . . . . . . . . . 21 |- ( p e. ( QQ X. QQ ) -> ( 1st ` p ) e. RR* )
46	45	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( QQ X. QQ ) ) -> ( 1st ` p ) e. RR* )
47		xp2nd 7199	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( p e. ( QQ X. QQ ) -> ( 2nd ` p ) e. QQ )
48	47	qred 39606	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( QQ X. QQ ) -> ( 2nd ` p ) e. RR )
49	48	rexrd 10089	. . . . . . . . . . . . . . . . . . . . 21 |- ( p e. ( QQ X. QQ ) -> ( 2nd ` p ) e. RR* )
50	49	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( QQ X. QQ ) ) -> ( 2nd ` p ) e. RR* )
51	41, 42, 7, 46, 50	smfpimioo 40994	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( QQ X. QQ ) ) -> ( `' F " ( ( 1st ` p ) (,) ( 2nd ` p ) ) ) e. ( S ↾t D ) )
52	40, 51	jca 554	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( QQ X. QQ ) ) -> ( ( ( 1st ` p ) (,) ( 2nd ` p ) ) e. ~P RR /\ ( `' F " ( ( 1st ` p ) (,) ( 2nd ` p ) ) ) e. ( S ↾t D ) ) )
53		imaeq2 5462	. . . . . . . . . . . . . . . . . . . 20 |- ( e = ( ( 1st ` p ) (,) ( 2nd ` p ) ) -> ( `' F " e ) = ( `' F " ( ( 1st ` p ) (,) ( 2nd ` p ) ) ) )
54	53	eleq1d 2686	. . . . . . . . . . . . . . . . . . 19 |- ( e = ( ( 1st ` p ) (,) ( 2nd ` p ) ) -> ( ( `' F " e ) e. ( S ↾t D ) <-> ( `' F " ( ( 1st ` p ) (,) ( 2nd ` p ) ) ) e. ( S ↾t D ) ) )
55	54, 8	elrab2 3366	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` p ) (,) ( 2nd ` p ) ) e. T <-> ( ( ( 1st ` p ) (,) ( 2nd ` p ) ) e. ~P RR /\ ( `' F " ( ( 1st ` p ) (,) ( 2nd ` p ) ) ) e. ( S ↾t D ) ) )
56	52, 55	sylibr 224	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( QQ X. QQ ) ) -> ( ( 1st ` p ) (,) ( 2nd ` p ) ) e. T )
57	56	3adant3 1081	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( QQ X. QQ ) /\ ( (,) ` p ) = q ) -> ( ( 1st ` p ) (,) ( 2nd ` p ) ) e. T )
58	35, 57	eqeltrd 2701	. . . . . . . . . . . . . . 15 |- ( ( ph /\ p e. ( QQ X. QQ ) /\ ( (,) ` p ) = q ) -> q e. T )
59	58	3exp 1264	. . . . . . . . . . . . . 14 |- ( ph -> ( p e. ( QQ X. QQ ) -> ( ( (,) ` p ) = q -> q e. T ) ) )
60	59	rexlimdv 3030	. . . . . . . . . . . . 13 |- ( ph -> ( E. p e. ( QQ X. QQ ) ( (,) ` p ) = q -> q e. T ) )
61	60	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ q e. ( (,) " ( QQ X. QQ ) ) ) -> ( E. p e. ( QQ X. QQ ) ( (,) ` p ) = q -> q e. T ) )
62	23, 61	mpd 15	. . . . . . . . . . 11 |- ( ( ph /\ q e. ( (,) " ( QQ X. QQ ) ) ) -> q e. T )
63	62	ssd 39252	. . . . . . . . . 10 |- ( ph -> ( (,) " ( QQ X. QQ ) ) C_ T )
64	63	adantr 481	. . . . . . . . 9 |- ( ( ph /\ q C_ ( (,) " ( QQ X. QQ ) ) ) -> ( (,) " ( QQ X. QQ ) ) C_ T )
65	17, 64	sstrd 3613	. . . . . . . 8 |- ( ( ph /\ q C_ ( (,) " ( QQ X. QQ ) ) ) -> q C_ T )
66	16, 65	elpwd 4167	. . . . . . 7 |- ( ( ph /\ q C_ ( (,) " ( QQ X. QQ ) ) ) -> q e. ~P T )
67		ssdomg 8001	. . . . . . . . . 10 |- ( ( (,) " ( QQ X. QQ ) ) e. _V -> ( q C_ ( (,) " ( QQ X. QQ ) ) -> q ~<_ ( (,) " ( QQ X. QQ ) ) ) )
68	12, 67	ax-mp 5	. . . . . . . . 9 |- ( q C_ ( (,) " ( QQ X. QQ ) ) -> q ~<_ ( (,) " ( QQ X. QQ ) ) )
69		qct 39578	. . . . . . . . . . . . 13 |- QQ ~<_ om
70	69, 69	pm3.2i 471	. . . . . . . . . . . 12 |- ( QQ ~<_ om /\ QQ ~<_ om )
71		xpct 8839	. . . . . . . . . . . 12 |- ( ( QQ ~<_ om /\ QQ ~<_ om ) -> ( QQ X. QQ ) ~<_ om )
72	70, 71	ax-mp 5	. . . . . . . . . . 11 |- ( QQ X. QQ ) ~<_ om
73		fimact 9357	. . . . . . . . . . 11 |- ( ( ( QQ X. QQ ) ~<_ om /\ Fun (,) ) -> ( (,) " ( QQ X. QQ ) ) ~<_ om )
74	72, 18, 73	mp2an 708	. . . . . . . . . 10 |- ( (,) " ( QQ X. QQ ) ) ~<_ om
75	74	a1i 11	. . . . . . . . 9 |- ( q C_ ( (,) " ( QQ X. QQ ) ) -> ( (,) " ( QQ X. QQ ) ) ~<_ om )
76		domtr 8009	. . . . . . . . 9 |- ( ( q ~<_ ( (,) " ( QQ X. QQ ) ) /\ ( (,) " ( QQ X. QQ ) ) ~<_ om ) -> q ~<_ om )
77	68, 75, 76	syl2anc 693	. . . . . . . 8 |- ( q C_ ( (,) " ( QQ X. QQ ) ) -> q ~<_ om )
78	77	adantl 482	. . . . . . 7 |- ( ( ph /\ q C_ ( (,) " ( QQ X. QQ ) ) ) -> q ~<_ om )
79	10, 66, 78	salunicl 40536	. . . . . 6 |- ( ( ph /\ q C_ ( (,) " ( QQ X. QQ ) ) ) -> U. q e. T )
80	79	adantrr 753	. . . . 5 |- ( ( ph /\ ( q C_ ( (,) " ( QQ X. QQ ) ) /\ G = U. q ) ) -> U. q e. T )
81	4, 80	eqeltrd 2701	. . . 4 |- ( ( ph /\ ( q C_ ( (,) " ( QQ X. QQ ) ) /\ G = U. q ) ) -> G e. T )
82	81	ex 450	. . 3 |- ( ph -> ( ( q C_ ( (,) " ( QQ X. QQ ) ) /\ G = U. q ) -> G e. T ) )
83	82	exlimdv 1861	. 2 |- ( ph -> ( E. q ( q C_ ( (,) " ( QQ X. QQ ) ) /\ G = U. q ) -> G e. T ) )
84	3, 83	mpd 15	1 |- ( ph -> G e. T )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   <.cop 4183   U.cuni 4436   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   omcom 7065   1stc1st 7166   2ndc2nd 7167   ~<_ cdom 7953   RRcr 9935   RR*cxr 10073   QQcq 11788   (,)cioo 12175   ↾_t crest 16081   topGenctg 16098  SAlgcsalg 40528  SMblFncsmblfn 40909

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-inf2 8538  ax-cc 9257  ax-ac2 9285  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-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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  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-rp 11833  df-ioo 12179  df-ico 12181  df-fl 12593  df-rest 16083  df-topgen 16104  df-bases 20750  df-salg 40529  df-smblfn 40910

This theorem is referenced by:  smfpimbor1lem2  41006