Theorem ustssel 22009

Description: A uniform structure is upward closed. Condition F_I of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.) (Proof shortened by AV, 17-Sep-2021.)

Assertion

Ref	Expression
ustssel	|- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> ( V C_ W -> W e. U ) )

Proof of Theorem ustssel

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

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> U e. (UnifOn ` X ) )
2	1	elfvexd 6222	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> X e. _V )
3		isust 22007	. . . . . 6 |- ( X e. _V -> ( U e. (UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
4	2, 3	syl 17	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> ( U e. (UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
5	1, 4	mpbid 222	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) )
6	5	simp3d 1075	. . 3 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) )
7		simp1 1061	. . . 4 |- ( ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) -> A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) )
8	7	ralimi 2952	. . 3 |- ( A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) -> A. v e. U A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) )
9	6, 8	syl 17	. 2 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> A. v e. U A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) )
10		simp2 1062	. . 3 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> V e. U )
11		xpexg 6960	. . . . 5 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
12	2, 2, 11	syl2anc 693	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> ( X X. X ) e. _V )
13		simp3 1063	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> W C_ ( X X. X ) )
14	12, 13	sselpwd 4807	. . 3 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> W e. ~P ( X X. X ) )
15		sseq1 3626	. . . . 5 |- ( v = V -> ( v C_ w <-> V C_ w ) )
16	15	imbi1d 331	. . . 4 |- ( v = V -> ( ( v C_ w -> w e. U ) <-> ( V C_ w -> w e. U ) ) )
17		sseq2 3627	. . . . 5 |- ( w = W -> ( V C_ w <-> V C_ W ) )
18		eleq1 2689	. . . . 5 |- ( w = W -> ( w e. U <-> W e. U ) )
19	17, 18	imbi12d 334	. . . 4 |- ( w = W -> ( ( V C_ w -> w e. U ) <-> ( V C_ W -> W e. U ) ) )
20	16, 19	rspc2v 3322	. . 3 |- ( ( V e. U /\ W e. ~P ( X X. X ) ) -> ( A. v e. U A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) -> ( V C_ W -> W e. U ) ) )
21	10, 14, 20	syl2anc 693	. 2 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> ( A. v e. U A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) -> ( V C_ W -> W e. U ) ) )
22	9, 21	mpd 15	1 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W C_ ( X X. X ) ) -> ( V C_ W -> W e. U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` 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-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-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-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-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ust 22004

This theorem is referenced by:  trust  22033  ustuqtop1  22045  ucnprima  22086