Theorem ustuqtop2 20480

Description: Lemma for ustuqtop 20484 (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 20445	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ w e. U /\ u e. U ) -> ( w i^i u ) e. U )
6	2, 3, 4, 5	syl3anc 1228	. . . . . . . . 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 3695	. . . . . . . . . . 11 |- ( ( a = ( w " { p } ) /\ b = ( u " { p } ) ) -> ( a i^i b ) = ( ( w " { p } ) i^i ( u " { p } ) ) )
9		vex 3116	. . . . . . . . . . . 12 |- p e. _V
10		inimasn 5421	. . . . . . . . . . . 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 2526	. . . . . . . . . 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 5330	. . . . . . . . . . 11 |- ( x = ( w i^i u ) -> ( x " { p } ) = ( ( w i^i u ) " { p } ) )
15	14	eqeq2d 2481	. . . . . . . . . 10 |- ( x = ( w i^i u ) -> ( ( a i^i b ) = ( x " { p } ) <-> ( a i^i b ) = ( ( w i^i u ) " { p } ) ) )
16	15	rspcev 3214	. . . . . . . . 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 3116	. . . . . . . . . . 11 |- a e. _V
19	18	inex1 4588	. . . . . . . . . 10 |- ( a i^i b ) e. _V
20		utopustuq.1	. . . . . . . . . . 11 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
21	20	ustuqtoplem 20477	. . . . . . . . . 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 3116	. . . . . . . . . 10 |- b e. _V
28	20	ustuqtoplem 20477	. . . . . . . . . 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 3003	. . . . . 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 20477	. . . . . . . . 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 3003	. . . . 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 2878	. . . 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 2878	. . 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 5874	. . . 4 |- ( N ` p ) e. _V
41		inficl 7881	. . . 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 3556	. 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 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   i^i cin 3475   C_ wss 3476   {csn 4027   |-> cmpt 4505   ran crn 5000   "cima 5002   ` cfv 5586   ficfi 7866  UnifOncust 20437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-fin 7517  df-fi 7867  df-ust 20438

This theorem is referenced by:  ustuqtop  20484  utopsnneiplem  20485