Theorem ustuqtop1 21187

Description: Lemma for ustuqtop 21192, similar to ssnei2 20063 (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 1056	. . . . . 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 1228	. . . . 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 760	. . . . 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 21150	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> w C_ ( X X. X ) )
5	2, 3, 4	syl2anc 665	. . . . . 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 1057	. . . . . . . . 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 1228	. . . . . . . 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 4148	. . . . . . 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 1010	. . . . . . . 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 1228	. . . . . . 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 4960	. . . . . . 7 |- ( ( { p } C_ X /\ b C_ X ) -> ( { p } X. b ) C_ ( X X. X ) )
12	8, 10, 11	syl2anc 665	. . . . . 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 3648	. . . . 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 3635	. . . . . 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 21151	. . . . . 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 430	. . . . 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 1267	. . . 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 1009	. . . . . 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 1228	. . . . 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 3642	. . . . . . 7 |- ( a C_ b <-> ( a u. b ) = b )
22	21	biimpi 197	. . . . . 6 |- ( a C_ b -> ( a u. b ) = b )
23		id 23	. . . . . . . 8 |- ( a = ( w " { p } ) -> a = ( w " { p } ) )
24		inidm 3677	. . . . . . . . . . 11 |- ( { p } i^i { p } ) = { p }
25		vex 3090	. . . . . . . . . . . 12 |- p e. _V
26	25	snnz 4121	. . . . . . . . . . 11 |- { p } =/= (/)
27	24, 26	eqnetri 2727	. . . . . . . . . 10 |- ( { p } i^i { p } ) =/= (/)
28		xpima2 5301	. . . . . . . . . 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 2437	. . . . . . . 8 |- ( a = ( w " { p } ) -> b = ( ( { p } X. b ) " { p } ) )
31	23, 30	uneq12d 3627	. . . . . . 7 |- ( a = ( w " { p } ) -> ( a u. b ) = ( ( w " { p } ) u. ( ( { p } X. b ) " { p } ) ) )
32		imaundir 5269	. . . . . . 7 |- ( ( w u. ( { p } X. b ) ) " { p } ) = ( ( w " { p } ) u. ( ( { p } X. b ) " { p } ) )
33	31, 32	syl6eqr 2488	. . . . . 6 |- ( a = ( w " { p } ) -> ( a u. b ) = ( ( w u. ( { p } X. b ) ) " { p } ) )
34	22, 33	sylan9req 2491	. . . . 5 |- ( ( a C_ b /\ a = ( w " { p } ) ) -> b = ( ( w u. ( { p } X. b ) ) " { p } ) )
35	20, 34	sylancom 671	. . . 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 5183	. . . . . 6 |- ( u = ( w u. ( { p } X. b ) ) -> ( u " { p } ) = ( ( w u. ( { p } X. b ) ) " { p } ) )
37	36	eqeq2d 2443	. . . . 5 |- ( u = ( w u. ( { p } X. b ) ) -> ( b = ( u " { p } ) <-> b = ( ( w u. ( { p } X. b ) ) " { p } ) ) )
38	37	rspcev 3188	. . . 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 665	. . 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 3090	. . . . . 6 |- a e. _V
41		utopustuq.1	. . . . . . 7 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
42	41	ustuqtoplem 21185	. . . . . 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 675	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
44	43	biimpa 486	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
45	44	3ad2antl1 1167	. . 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 2977	. 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 3090	. . . . 5 |- b e. _V
48	41	ustuqtoplem 21185	. . . . 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 675	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
50	49	3ad2ant1 1026	. . 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 466	. 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 235	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 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   =/= wne 2625   E.wrex 2783   _Vcvv 3087   u. cun 3440   i^i cin 3441   C_ wss 3442   (/)c0 3767   {csn 4002   |-> cmpt 4484   X. cxp 4852   ran crn 4855   "cima 4857   ` cfv 5601  UnifOncust 21145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ust 21146

This theorem is referenced by:  ustuqtop4  21190  ustuqtop  21192  utopsnneiplem  21193