Theorem ustuqtop3 22047

Description: Lemma for ustuqtop 22050, similar to elnei 20915. (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
ustuqtop3	|- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> p e. a )

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

Allowed substitution hint:   N( v)

Proof of Theorem ustuqtop3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnresi 6008	. . . . . . 7 |- ( _I |` X ) Fn X
2		fnsnfv 6258	. . . . . . 7 |- ( ( ( _I |` X ) Fn X /\ p e. X ) -> { ( ( _I |` X ) ` p ) } = ( ( _I |` X ) " { p } ) )
3	1, 2	mpan 706	. . . . . 6 |- ( p e. X -> { ( ( _I |` X ) ` p ) } = ( ( _I |` X ) " { p } ) )
4	3	ad4antlr 769	. . . . 5 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> { ( ( _I |` X ) ` p ) } = ( ( _I |` X ) " { p } ) )
5		simp-4l 806	. . . . . . 7 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> U e. (UnifOn ` X ) )
6		simplr 792	. . . . . . 7 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> w e. U )
7		ustdiag 22012	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> ( _I |` X ) C_ w )
8	5, 6, 7	syl2anc 693	. . . . . 6 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( _I |` X ) C_ w )
9		imass1 5500	. . . . . 6 |- ( ( _I |` X ) C_ w -> ( ( _I |` X ) " { p } ) C_ ( w " { p } ) )
10	8, 9	syl 17	. . . . 5 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( ( _I |` X ) " { p } ) C_ ( w " { p } ) )
11	4, 10	eqsstrd 3639	. . . 4 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> { ( ( _I |` X ) ` p ) } C_ ( w " { p } ) )
12		fvex 6201	. . . . 5 |- ( ( _I |` X ) ` p ) e. _V
13	12	snss 4316	. . . 4 |- ( ( ( _I |` X ) ` p ) e. ( w " { p } ) <-> { ( ( _I |` X ) ` p ) } C_ ( w " { p } ) )
14	11, 13	sylibr 224	. . 3 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( ( _I |` X ) ` p ) e. ( w " { p } ) )
15		fvresi 6439	. . . . 5 |- ( p e. X -> ( ( _I |` X ) ` p ) = p )
16	15	eqcomd 2628	. . . 4 |- ( p e. X -> p = ( ( _I |` X ) ` p ) )
17	16	ad4antlr 769	. . 3 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> p = ( ( _I |` X ) ` p ) )
18		simpr 477	. . 3 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> a = ( w " { p } ) )
19	14, 17, 18	3eltr4d 2716	. 2 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> p e. a )
20		vex 3203	. . . 4 |- a e. _V
21		utopustuq.1	. . . . 5 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
22	21	ustuqtoplem 22043	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. _V ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
23	20, 22	mpan2 707	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
24	23	biimpa 501	. 2 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
25	19, 24	r19.29a 3078	1 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> p e. a )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {csn 4177   |-> cmpt 4729   _I cid 5023   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   ` 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:  ustuqtop  22050  utopsnneiplem  22051