Theorem ustuqtop2 19797

Description: Lemma for ustuqtop 19801 (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

Ref	Expression
utopustuq.1	|- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )

Assertion

Ref	Expression
ustuqtop2	|- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( fi ` ( N ` p ) ) C_ ( N ` p ) )

Distinct variable groups:   v, p, U   X, p, v   N, p

Allowed substitution hint:   N( v)

Proof of Theorem ustuqtop2

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

Step	Hyp	Ref	Expression
1		simp-6l 769	. . . . . . . 8 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) /\ u e. U ) /\ b = ( u " { p } ) ) -> ( U e. (UnifOn ` X ) /\ p e. X ) )
2		simp-7l 771	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) /\ u e. U ) /\ b = ( u " { p } ) ) -> U e. (UnifOn ` X ) )
3		simp-4r 766	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) /\ u e. U ) /\ b = ( u " { p } ) ) -> w e. U )
4		simplr 754	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) /\ u e. U ) /\ b = ( u " { p } ) ) -> u e. U )
5		ustincl 19762	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ w e. U /\ u e. U ) -> ( w i^i u ) e. U )
6	2, 3, 4, 5	syl3anc 1218	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) /\ u e. U ) /\ b = ( u " { p } ) ) -> ( w i^i u ) e. U )
7		simpllr 758	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) /\ u e. U ) /\ b = ( u " { p } ) ) -> a = ( w " { p } ) )
8		ineq12 3542	. . . . . . . . . . 11 |- ( ( a = ( w " { p } ) /\ b = ( u " { p } ) ) -> ( a i^i b ) = ( ( w " { p } ) i^i ( u " { p } ) ) )
9		vex 2970	. . . . . . . . . . . 12 |- p e. _V
10		inimasn 5249	. . . . . . . . . . . 12 |- ( p e. _V -> ( ( w i^i u ) " { p } ) = ( ( w " { p } ) i^i ( u " { p } ) ) )
11	9, 10	ax-mp 5	. . . . . . . . . . 11 |- ( ( w i^i u ) " { p } ) = ( ( w " { p } ) i^i ( u " { p } ) )
12	8, 11	syl6eqr 2488	. . . . . . . . . 10 |- ( ( a = ( w " { p } ) /\ b = ( u " { p } ) ) -> ( a i^i b ) = ( ( w i^i u ) " { p } ) )
13	7, 12	sylancom 667	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) /\ u e. U ) /\ b = ( u " { p } ) ) -> ( a i^i b ) = ( ( w i^i u ) " { p } ) )
14		imaeq1 5159	. . . . . . . . . . 11 |- ( x = ( w i^i u ) -> ( x " { p } ) = ( ( w i^i u ) " { p } ) )
15	14	eqeq2d 2449	. . . . . . . . . 10 |- ( x = ( w i^i u ) -> ( ( a i^i b ) = ( x " { p } ) <-> ( a i^i b ) = ( ( w i^i u ) " { p } ) ) )
16	15	rspcev 3068	. . . . . . . . 9 |- ( ( ( w i^i u ) e. U /\ ( a i^i b ) = ( ( w i^i u ) " { p } ) ) -> E. x e. U ( a i^i b ) = ( x " { p } ) )
17	6, 13, 16	syl2anc 661	. . . . . . . 8 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) /\ u e. U ) /\ b = ( u " { p } ) ) -> E. x e. U ( a i^i b ) = ( x " { p } ) )
18		vex 2970	. . . . . . . . . . 11 |- a e. _V
19	18	inex1 4428	. . . . . . . . . 10 |- ( a i^i b ) e. _V
20		utopustuq.1	. . . . . . . . . . 11 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
21	20	ustuqtoplem 19794	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ ( a i^i b ) e. _V ) -> ( ( a i^i b ) e. ( N ` p ) <-> E. x e. U ( a i^i b ) = ( x " { p } ) ) )
22	19, 21	mpan2 671	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( ( a i^i b ) e. ( N ` p ) <-> E. x e. U ( a i^i b ) = ( x " { p } ) ) )
23	22	biimpar 485	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ E. x e. U ( a i^i b ) = ( x " { p } ) ) -> ( a i^i b ) e. ( N ` p ) )
24	1, 17, 23	syl2anc 661	. . . . . . 7 |- ( ( ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) /\ u e. U ) /\ b = ( u " { p } ) ) -> ( a i^i b ) e. ( N ` p ) )
25		simp-4l 765	. . . . . . . 8 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( U e. (UnifOn ` X ) /\ p e. X ) )
26		simpllr 758	. . . . . . . 8 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> b e. ( N ` p ) )
27		vex 2970	. . . . . . . . . 10 |- b e. _V
28	20	ustuqtoplem 19794	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ b e. _V ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
29	27, 28	mpan2 671	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
30	29	biimpa 484	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ b e. ( N ` p ) ) -> E. u e. U b = ( u " { p } ) )
31	25, 26, 30	syl2anc 661	. . . . . . 7 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> E. u e. U b = ( u " { p } ) )
32	24, 31	r19.29a 2857	. . . . . 6 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( a i^i b ) e. ( N ` p ) )
33	20	ustuqtoplem 19794	. . . . . . . . 9 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. _V ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
34	18, 33	mpan2 671	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
35	34	biimpa 484	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
36	35	adantr 465	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
37	32, 36	r19.29a 2857	. . . . 5 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ b e. ( N ` p ) ) -> ( a i^i b ) e. ( N ` p ) )
38	37	ralrimiva 2794	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> A. b e. ( N ` p ) ( a i^i b ) e. ( N ` p ) )
39	38	ralrimiva 2794	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> A. a e. ( N ` p ) A. b e. ( N ` p ) ( a i^i b ) e. ( N ` p ) )
40		fvex 5696	. . . 4 |- ( N ` p ) e. _V
41		inficl 7667	. . . 4 |- ( ( N ` p ) e. _V -> ( A. a e. ( N ` p ) A. b e. ( N ` p ) ( a i^i b ) e. ( N ` p ) <-> ( fi ` ( N ` p ) ) = ( N ` p ) ) )
42	40, 41	ax-mp 5	. . 3 |- ( A. a e. ( N ` p ) A. b e. ( N ` p ) ( a i^i b ) e. ( N ` p ) <-> ( fi ` ( N ` p ) ) = ( N ` p ) )
43	39, 42	sylib 196	. 2 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( fi ` ( N ` p ) ) = ( N ` p ) )
44		eqimss 3403	. 2 |- ( ( fi ` ( N ` p ) ) = ( N ` p ) -> ( fi ` ( N ` p ) ) C_ ( N ` p ) )
45	43, 44	syl 16	1 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( fi ` ( N ` p ) ) C_ ( N ` p ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2710   E.wrex 2711   _Vcvv 2967   i^i cin 3322   C_ wss 3323   {csn 3872   e. cmpt 4345   ran crn 4836   "cima 4838   ` cfv 5413   ficfi 7652  UnifOncust 19754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-fin 7306  df-fi 7653  df-ust 19755

This theorem is referenced by:  ustuqtop  19801  utopsnneiplem  19802