Theorem ustuqtop1 20495

Description: Lemma for ustuqtop 20500, similar to ssnei2 19399 (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 1047	. . . . . 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 1218	. . . . 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 754	. . . . 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 20458	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> w C_ ( X X. X ) )
5	2, 3, 4	syl2anc 661	. . . . . 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 1048	. . . . . . . . 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 1218	. . . . . . . 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 4172	. . . . . . 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 1001	. . . . . . . 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 1218	. . . . . . 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 5107	. . . . . . 7 |- ( ( { p } C_ X /\ b C_ X ) -> ( { p } X. b ) C_ ( X X. X ) )
12	8, 10, 11	syl2anc 661	. . . . . 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 3680	. . . . 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 3667	. . . . . 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 20459	. . . . . 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 429	. . . . 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 1231	. . . 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 1000	. . . . . 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 1218	. . . . 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 3674	. . . . . . 7 |- ( a C_ b <-> ( a u. b ) = b )
22	21	biimpi 194	. . . . . 6 |- ( a C_ b -> ( a u. b ) = b )
23		id 22	. . . . . . . 8 |- ( a = ( w " { p } ) -> a = ( w " { p } ) )
24		inidm 3707	. . . . . . . . . . 11 |- ( { p } i^i { p } ) = { p }
25		vex 3116	. . . . . . . . . . . 12 |- p e. _V
26	25	snnz 4145	. . . . . . . . . . 11 |- { p } =/= (/)
27	24, 26	eqnetri 2763	. . . . . . . . . 10 |- ( { p } i^i { p } ) =/= (/)
28		xpima2 5450	. . . . . . . . . 10 |- ( ( { p } i^i { p } ) =/= (/) -> ( ( { p } X. b ) " { p } ) = b )
29	27, 28	mp1i 12	. . . . . . . . 9 |- ( a = ( w " { p } ) -> ( ( { p } X. b ) " { p } ) = b )
30	29	eqcomd 2475	. . . . . . . 8 |- ( a = ( w " { p } ) -> b = ( ( { p } X. b ) " { p } ) )
31	23, 30	uneq12d 3659	. . . . . . 7 |- ( a = ( w " { p } ) -> ( a u. b ) = ( ( w " { p } ) u. ( ( { p } X. b ) " { p } ) ) )
32		imaundir 5418	. . . . . . 7 |- ( ( w u. ( { p } X. b ) ) " { p } ) = ( ( w " { p } ) u. ( ( { p } X. b ) " { p } ) )
33	31, 32	syl6eqr 2526	. . . . . 6 |- ( a = ( w " { p } ) -> ( a u. b ) = ( ( w u. ( { p } X. b ) ) " { p } ) )
34	22, 33	sylan9req 2529	. . . . 5 |- ( ( a C_ b /\ a = ( w " { p } ) ) -> b = ( ( w u. ( { p } X. b ) ) " { p } ) )
35	20, 34	sylancom 667	. . . 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 5331	. . . . . 6 |- ( u = ( w u. ( { p } X. b ) ) -> ( u " { p } ) = ( ( w u. ( { p } X. b ) ) " { p } ) )
37	36	eqeq2d 2481	. . . . 5 |- ( u = ( w u. ( { p } X. b ) ) -> ( b = ( u " { p } ) <-> b = ( ( w u. ( { p } X. b ) ) " { p } ) ) )
38	37	rspcev 3214	. . . 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 661	. . 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 3116	. . . . . 6 |- a e. _V
41		utopustuq.1	. . . . . . 7 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
42	41	ustuqtoplem 20493	. . . . . 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 671	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
44	43	biimpa 484	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
45	44	3ad2antl1 1158	. . 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 3003	. 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 3116	. . . . 5 |- b e. _V
48	41	ustuqtoplem 20493	. . . . 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 671	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
50	49	3ad2ant1 1017	. . 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 465	. 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 232	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 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   E.wrex 2815   _Vcvv 3113   u. cun 3474   i^i cin 3475   C_ wss 3476   (/)c0 3785   {csn 4027   |-> cmpt 4505   X. cxp 4997   ran crn 5000   "cima 5002   ` cfv 5587  UnifOncust 20453

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 6575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ust 20454

This theorem is referenced by:  ustuqtop4  20498  ustuqtop  20500  utopsnneiplem  20501