Theorem ustuqtop1 21256

Description: Lemma for ustuqtop 21261, similar to ssnei2 20132. (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 1059	. . . . . 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 1232	. . . . 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 762	. . . . 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 21219	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> w C_ ( X X. X ) )
5	2, 3, 4	syl2anc 667	. . . . . 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 1060	. . . . . . . . 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 1232	. . . . . . . 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 4117	. . . . . . 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 1013	. . . . . . . 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 1232	. . . . . . 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 4940	. . . . . . 7 |- ( ( { p } C_ X /\ b C_ X ) -> ( { p } X. b ) C_ ( X X. X ) )
12	8, 10, 11	syl2anc 667	. . . . . 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 3610	. . . . 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 3597	. . . . . 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 21220	. . . . . 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 431	. . . . 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 1271	. . . 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 1012	. . . . . 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 1232	. . . . 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 3604	. . . . . . 7 |- ( a C_ b <-> ( a u. b ) = b )
22	21	biimpi 198	. . . . . 6 |- ( a C_ b -> ( a u. b ) = b )
23		id 22	. . . . . . . 8 |- ( a = ( w " { p } ) -> a = ( w " { p } ) )
24		inidm 3641	. . . . . . . . . . 11 |- ( { p } i^i { p } ) = { p }
25		vex 3048	. . . . . . . . . . . 12 |- p e. _V
26	25	snnz 4090	. . . . . . . . . . 11 |- { p } =/= (/)
27	24, 26	eqnetri 2694	. . . . . . . . . 10 |- ( { p } i^i { p } ) =/= (/)
28		xpima2 5281	. . . . . . . . . 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 2457	. . . . . . . 8 |- ( a = ( w " { p } ) -> b = ( ( { p } X. b ) " { p } ) )
31	23, 30	uneq12d 3589	. . . . . . 7 |- ( a = ( w " { p } ) -> ( a u. b ) = ( ( w " { p } ) u. ( ( { p } X. b ) " { p } ) ) )
32		imaundir 5249	. . . . . . 7 |- ( ( w u. ( { p } X. b ) ) " { p } ) = ( ( w " { p } ) u. ( ( { p } X. b ) " { p } ) )
33	31, 32	syl6eqr 2503	. . . . . 6 |- ( a = ( w " { p } ) -> ( a u. b ) = ( ( w u. ( { p } X. b ) ) " { p } ) )
34	22, 33	sylan9req 2506	. . . . 5 |- ( ( a C_ b /\ a = ( w " { p } ) ) -> b = ( ( w u. ( { p } X. b ) ) " { p } ) )
35	20, 34	sylancom 673	. . . 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 5163	. . . . . 6 |- ( u = ( w u. ( { p } X. b ) ) -> ( u " { p } ) = ( ( w u. ( { p } X. b ) ) " { p } ) )
37	36	eqeq2d 2461	. . . . 5 |- ( u = ( w u. ( { p } X. b ) ) -> ( b = ( u " { p } ) <-> b = ( ( w u. ( { p } X. b ) ) " { p } ) ) )
38	37	rspcev 3150	. . . 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 667	. . 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 3048	. . . . . 6 |- a e. _V
41		utopustuq.1	. . . . . . 7 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
42	41	ustuqtoplem 21254	. . . . . 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 677	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
44	43	biimpa 487	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
45	44	3ad2antl1 1170	. . 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 2932	. 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 3048	. . . . 5 |- b e. _V
48	41	ustuqtoplem 21254	. . . . 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 677	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( b e. ( N ` p ) <-> E. u e. U b = ( u " { p } ) ) )
50	49	3ad2ant1 1029	. . 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 467	. 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 236	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   E.wrex 2738   _Vcvv 3045   u. cun 3402   i^i cin 3403   C_ wss 3404   (/)c0 3731   {csn 3968   |-> cmpt 4461   X. cxp 4832   ran crn 4835   "cima 4837   ` cfv 5582  UnifOncust 21214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ust 21215

This theorem is referenced by:  ustuqtop4  21259  ustuqtop  21261  utopsnneiplem  21262