Theorem ustuqtop2 22046

Description: Lemma for ustuqtop 22050. (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 810	. . . . . . . 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 812	. . . . . . . . . 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 807	. . . . . . . . . 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 792	. . . . . . . . . 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 22011	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ w e. U /\ u e. U ) -> ( w i^i u ) e. U )
6	2, 3, 4, 5	syl3anc 1326	. . . . . . . . 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 799	. . . . . . . . . 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 3809	. . . . . . . . . . 11 |- ( ( a = ( w " { p } ) /\ b = ( u " { p } ) ) -> ( a i^i b ) = ( ( w " { p } ) i^i ( u " { p } ) ) )
9		vex 3203	. . . . . . . . . . . 12 |- p e. _V
10		inimasn 5550	. . . . . . . . . . . 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 2674	. . . . . . . . . 10 |- ( ( a = ( w " { p } ) /\ b = ( u " { p } ) ) -> ( a i^i b ) = ( ( w i^i u ) " { p } ) )
13	7, 12	sylancom 701	. . . . . . . . 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 5461	. . . . . . . . . . 11 |- ( x = ( w i^i u ) -> ( x " { p } ) = ( ( w i^i u ) " { p } ) )
15	14	eqeq2d 2632	. . . . . . . . . 10 |- ( x = ( w i^i u ) -> ( ( a i^i b ) = ( x " { p } ) <-> ( a i^i b ) = ( ( w i^i u ) " { p } ) ) )
16	15	rspcev 3309	. . . . . . . . 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 693	. . . . . . . 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 3203	. . . . . . . . . . 11 |- a e. _V
19	18	inex1 4799	. . . . . . . . . 10 |- ( a i^i b ) e. _V
20		utopustuq.1	. . . . . . . . . . 11 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
21	20	ustuqtoplem 22043	. . . . . . . . . 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 707	. . . . . . . . 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 502	. . . . . . . 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 693	. . . . . . 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 806	. . . . . . . 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 799	. . . . . . . 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 3203	. . . . . . . . . 10 |- b e. _V
28	20	ustuqtoplem 22043	. . . . . . . . . 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 707	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
30	29	biimpa 501	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ b e. ( N ` p ) ) -> E. u e. U b = ( u " { p } ) )
31	25, 26, 30	syl2anc 693	. . . . . . 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 3078	. . . . . 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 22043	. . . . . . . . 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 707	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
35	34	biimpa 501	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
36	35	adantr 481	. . . . . 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 3078	. . . . 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 2966	. . . 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 2966	. . 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 6201	. . . 4 |- ( N ` p ) e. _V
41		inficl 8331	. . . 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 208	. 2 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( fi ` ( N ` p ) ) = ( N ` p ) )
44		eqimss 3657	. 2 |- ( ( fi ` ( N ` p ) ) = ( N ` p ) -> ( fi ` ( N ` p ) ) C_ ( N ` p ) )
45	43, 44	syl 17	1 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( fi ` ( N ` p ) ) C_ ( N ` p ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   |-> cmpt 4729   ran crn 5115   "cima 5117   ` cfv 5888   ficfi 8316  UnifOncust 22003

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

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-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-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-fin 7959  df-fi 8317  df-ust 22004

This theorem is referenced by:  ustuqtop  22050  utopsnneiplem  22051