Theorem ustuqtop2 18225

Description: Lemma for ustuqtop 18229 (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 747	. . . . . . . 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 749	. . . . . . . . . 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 744	. . . . . . . . . 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 732	. . . . . . . . . 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 18190	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ w e. U /\ u e. U ) -> ( w i^i u ) e. U )
6	2, 3, 4, 5	syl3anc 1184	. . . . . . . . 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 736	. . . . . . . . . 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 3497	. . . . . . . . . . 11 |- ( ( a = ( w " { p } ) /\ b = ( u " { p } ) ) -> ( a i^i b ) = ( ( w " { p } ) i^i ( u " { p } ) ) )
9		vex 2919	. . . . . . . . . . . 12 |- p e. _V
10		inimasn 5248	. . . . . . . . . . . 12 |- ( p e. _V -> ( ( w i^i u ) " { p } ) = ( ( w " { p } ) i^i ( u " { p } ) ) )
11	9, 10	ax-mp 8	. . . . . . . . . . 11 |- ( ( w i^i u ) " { p } ) = ( ( w " { p } ) i^i ( u " { p } ) )
12	8, 11	syl6eqr 2454	. . . . . . . . . 10 |- ( ( a = ( w " { p } ) /\ b = ( u " { p } ) ) -> ( a i^i b ) = ( ( w i^i u ) " { p } ) )
13	7, 12	sylancom 649	. . . . . . . . 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 5157	. . . . . . . . . . 11 |- ( x = ( w i^i u ) -> ( x " { p } ) = ( ( w i^i u ) " { p } ) )
15	14	eqeq2d 2415	. . . . . . . . . 10 |- ( x = ( w i^i u ) -> ( ( a i^i b ) = ( x " { p } ) <-> ( a i^i b ) = ( ( w i^i u ) " { p } ) ) )
16	15	rspcev 3012	. . . . . . . . 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 643	. . . . . . . 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 2919	. . . . . . . . . . 11 |- a e. _V
19	18	inex1 4304	. . . . . . . . . 10 |- ( a i^i b ) e. _V
20		utopustuq.1	. . . . . . . . . . 11 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
21	20	ustuqtoplem 18222	. . . . . . . . . 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 653	. . . . . . . . 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 472	. . . . . . . 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 643	. . . . . . 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 743	. . . . . . . 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 736	. . . . . . . 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 2919	. . . . . . . . . 10 |- b e. _V
28	20	ustuqtoplem 18222	. . . . . . . . . 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 653	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
30	29	biimpa 471	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ b e. ( N ` p ) ) -> E. u e. U b = ( u " { p } ) )
31	25, 26, 30	syl2anc 643	. . . . . . 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 2810	. . . . . 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 18222	. . . . . . . . 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 653	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
35	34	biimpa 471	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
36	35	adantr 452	. . . . . 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 2810	. . . . 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 2749	. . . 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 2749	. . 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 5701	. . . 4 |- ( N ` p ) e. _V
41		inficl 7388	. . . 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 8	. . 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 189	. 2 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( fi ` ( N ` p ) ) = ( N ` p ) )
44		eqimss 3360	. 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 177   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916   i^i cin 3279   C_ wss 3280   {csn 3774   e. cmpt 4226   ran crn 4838   "cima 4840   ` cfv 5413   ficfi 7373  UnifOncust 18182

This theorem is referenced by:  ustuqtop  18229  utopsnneiplem  18230

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-fin 7072  df-fi 7374  df-ust 18183