Theorem ustuqtop3 20474

Description: Lemma for ustuqtop 20477, similar to elnei 19371 (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 5689	. . . . . . 7 |- ( _I |` X ) Fn X
2		fnsnfv 5918	. . . . . . 7 |- ( ( ( _I |` X ) Fn X /\ p e. X ) -> { ( ( _I |` X ) ` p ) } = ( ( _I |` X ) " { p } ) )
3	1, 2	mpan 670	. . . . . 6 |- ( p e. X -> { ( ( _I |` X ) ` p ) } = ( ( _I |` X ) " { p } ) )
4	3	ad4antlr 732	. . . . 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 765	. . . . . . 7 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> U e. (UnifOn ` X ) )
6		simplr 754	. . . . . . 7 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> w e. U )
7		ustdiag 20439	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> ( _I |` X ) C_ w )
8	5, 6, 7	syl2anc 661	. . . . . 6 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( _I |` X ) C_ w )
9		imass1 5362	. . . . . 6 |- ( ( _I |` X ) C_ w -> ( ( _I |` X ) " { p } ) C_ ( w " { p } ) )
10	8, 9	syl 16	. . . . 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 3531	. . . 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 5867	. . . . 5 |- ( ( _I |` X ) ` p ) e. _V
13	12	snss 4144	. . . 4 |- ( ( ( _I |` X ) ` p ) e. ( w " { p } ) <-> { ( ( _I |` X ) ` p ) } C_ ( w " { p } ) )
14	11, 13	sylibr 212	. . 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 6078	. . . . 5 |- ( p e. X -> ( ( _I |` X ) ` p ) = p )
16	15	eqcomd 2468	. . . 4 |- ( p e. X -> p = ( ( _I |` X ) ` p ) )
17	16	ad4antlr 732	. . 3 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> p = ( ( _I |` X ) ` p ) )
18		simpr 461	. . 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 2563	. 2 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> p e. a )
20		vex 3109	. . . 4 |- a e. _V
21		utopustuq.1	. . . . 5 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
22	21	ustuqtoplem 20470	. . . 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 671	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
24	23	biimpa 484	. 2 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
25	19, 24	r19.29a 2996	1 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> p e. a )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   e. wcel 1762   E.wrex 2808   _Vcvv 3106   C_ wss 3469   {csn 4020   |-> cmpt 4498   _I cid 4783   ran crn 4993   |` cres 4994   "cima 4995   Fn wfn 5574   ` cfv 5579  UnifOncust 20430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ust 20431

This theorem is referenced by:  ustuqtop  20477  utopsnneiplem  20478