Theorem ustuqtop1 22045

Description: Lemma for ustuqtop 22050, similar to ssnei2 20920. (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
ustuqtop1	|- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) -> b e. ( N ` p ) )

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

Allowed substitution hint:   N( v)

Proof of Theorem ustuqtop1

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

Step	Hyp	Ref	Expression
1		simpl1l 1112	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ ( a e. ( N ` p ) /\ w e. U /\ a = ( w " { p } ) ) ) -> U e. (UnifOn ` X ) )
2	1	3anassrs 1290	. . . . 5 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> U e. (UnifOn ` X ) )
3		simplr 792	. . . . 5 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> w e. U )
4		ustssxp 22008	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> w C_ ( X X. X ) )
5	2, 3, 4	syl2anc 693	. . . . . 6 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> w C_ ( X X. X ) )
6		simpl1r 1113	. . . . . . . . 9 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ ( a e. ( N ` p ) /\ w e. U /\ a = ( w " { p } ) ) ) -> p e. X )
7	6	3anassrs 1290	. . . . . . . 8 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> p e. X )
8	7	snssd 4340	. . . . . . 7 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> { p } C_ X )
9		simpl3 1066	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ ( a e. ( N ` p ) /\ w e. U /\ a = ( w " { p } ) ) ) -> b C_ X )
10	9	3anassrs 1290	. . . . . . 7 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> b C_ X )
11		xpss12 5225	. . . . . . 7 |- ( ( { p } C_ X /\ b C_ X ) -> ( { p } X. b ) C_ ( X X. X ) )
12	8, 10, 11	syl2anc 693	. . . . . 6 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( { p } X. b ) C_ ( X X. X ) )
13	5, 12	unssd 3789	. . . . 5 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( w u. ( { p } X. b ) ) C_ ( X X. X ) )
14		ssun1 3776	. . . . . 6 |- w C_ ( w u. ( { p } X. b ) )
15	14	a1i 11	. . . . 5 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> w C_ ( w u. ( { p } X. b ) ) )
16		ustssel 22009	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ w e. U /\ ( w u. ( { p } X. b ) ) C_ ( X X. X ) ) -> ( w C_ ( w u. ( { p } X. b ) ) -> ( w u. ( { p } X. b ) ) e. U ) )
17	16	imp 445	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ w e. U /\ ( w u. ( { p } X. b ) ) C_ ( X X. X ) ) /\ w C_ ( w u. ( { p } X. b ) ) ) -> ( w u. ( { p } X. b ) ) e. U )
18	2, 3, 13, 15, 17	syl31anc 1329	. . . 4 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( w u. ( { p } X. b ) ) e. U )
19		simpl2 1065	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ ( a e. ( N ` p ) /\ w e. U /\ a = ( w " { p } ) ) ) -> a C_ b )
20	19	3anassrs 1290	. . . . 5 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> a C_ b )
21		ssequn1 3783	. . . . . . 7 |- ( a C_ b <-> ( a u. b ) = b )
22	21	biimpi 206	. . . . . 6 |- ( a C_ b -> ( a u. b ) = b )
23		id 22	. . . . . . . 8 |- ( a = ( w " { p } ) -> a = ( w " { p } ) )
24		inidm 3822	. . . . . . . . . . 11 |- ( { p } i^i { p } ) = { p }
25		vex 3203	. . . . . . . . . . . 12 |- p e. _V
26	25	snnz 4309	. . . . . . . . . . 11 |- { p } =/= (/)
27	24, 26	eqnetri 2864	. . . . . . . . . 10 |- ( { p } i^i { p } ) =/= (/)
28		xpima2 5578	. . . . . . . . . 10 |- ( ( { p } i^i { p } ) =/= (/) -> ( ( { p } X. b ) " { p } ) = b )
29	27, 28	mp1i 13	. . . . . . . . 9 |- ( a = ( w " { p } ) -> ( ( { p } X. b ) " { p } ) = b )
30	29	eqcomd 2628	. . . . . . . 8 |- ( a = ( w " { p } ) -> b = ( ( { p } X. b ) " { p } ) )
31	23, 30	uneq12d 3768	. . . . . . 7 |- ( a = ( w " { p } ) -> ( a u. b ) = ( ( w " { p } ) u. ( ( { p } X. b ) " { p } ) ) )
32		imaundir 5546	. . . . . . 7 |- ( ( w u. ( { p } X. b ) ) " { p } ) = ( ( w " { p } ) u. ( ( { p } X. b ) " { p } ) )
33	31, 32	syl6eqr 2674	. . . . . 6 |- ( a = ( w " { p } ) -> ( a u. b ) = ( ( w u. ( { p } X. b ) ) " { p } ) )
34	22, 33	sylan9req 2677	. . . . 5 |- ( ( a C_ b /\ a = ( w " { p } ) ) -> b = ( ( w u. ( { p } X. b ) ) " { p } ) )
35	20, 34	sylancom 701	. . . 4 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> b = ( ( w u. ( { p } X. b ) ) " { p } ) )
36		imaeq1 5461	. . . . . 6 |- ( u = ( w u. ( { p } X. b ) ) -> ( u " { p } ) = ( ( w u. ( { p } X. b ) ) " { p } ) )
37	36	eqeq2d 2632	. . . . 5 |- ( u = ( w u. ( { p } X. b ) ) -> ( b = ( u " { p } ) <-> b = ( ( w u. ( { p } X. b ) ) " { p } ) ) )
38	37	rspcev 3309	. . . 4 |- ( ( ( w u. ( { p } X. b ) ) e. U /\ b = ( ( w u. ( { p } X. b ) ) " { p } ) ) -> E. u e. U b = ( u " { p } ) )
39	18, 35, 38	syl2anc 693	. . 3 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> E. u e. U b = ( u " { p } ) )
40		vex 3203	. . . . . 6 |- a e. _V
41		utopustuq.1	. . . . . . 7 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
42	41	ustuqtoplem 22043	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. _V ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
43	40, 42	mpan2 707	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
44	43	biimpa 501	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
45	44	3ad2antl1 1223	. . 3 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
46	39, 45	r19.29a 3078	. 2 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) -> E. u e. U b = ( u " { p } ) )
47		vex 3203	. . . . 5 |- b e. _V
48	41	ustuqtoplem 22043	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ b e. _V ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
49	47, 48	mpan2 707	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
50	49	3ad2ant1 1082	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
51	50	adantr 481	. 2 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
52	46, 51	mpbird 247	1 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) -> b e. ( N ` p ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. cxp 5112   ran crn 5115   "cima 5117   ` cfv 5888  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ust 22004

This theorem is referenced by:  ustuqtop4  22048  ustuqtop  22050  utopsnneiplem  22051