Theorem ustuqtop3 21313

Description: Lemma for ustuqtop 21316, similar to elnei 20182. (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 5719	. . . . . . 7 |- ( _I |` X ) Fn X
2		fnsnfv 5953	. . . . . . 7 |- ( ( ( _I |` X ) Fn X /\ p e. X ) -> { ( ( _I |` X ) ` p ) } = ( ( _I |` X ) " { p } ) )
3	1, 2	mpan 681	. . . . . 6 |- ( p e. X -> { ( ( _I |` X ) ` p ) } = ( ( _I |` X ) " { p } ) )
4	3	ad4antlr 744	. . . . 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 781	. . . . . . 7 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> U e. (UnifOn ` X ) )
6		simplr 767	. . . . . . 7 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> w e. U )
7		ustdiag 21278	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> ( _I |` X ) C_ w )
8	5, 6, 7	syl2anc 671	. . . . . 6 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> ( _I |` X ) C_ w )
9		imass1 5225	. . . . . 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 3478	. . . 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 5902	. . . . 5 |- ( ( _I |` X ) ` p ) e. _V
13	12	snss 4109	. . . 4 |- ( ( ( _I |` X ) ` p ) e. ( w " { p } ) <-> { ( ( _I |` X ) ` p ) } C_ ( w " { p } ) )
14	11, 13	sylibr 217	. . 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 6119	. . . . 5 |- ( p e. X -> ( ( _I |` X ) ` p ) = p )
16	15	eqcomd 2468	. . . 4 |- ( p e. X -> p = ( ( _I |` X ) ` p ) )
17	16	ad4antlr 744	. . 3 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> p = ( ( _I |` X ) ` p ) )
18		simpr 467	. . 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 2555	. 2 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> p e. a )
20		vex 3060	. . . 4 |- a e. _V
21		utopustuq.1	. . . . 5 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
22	21	ustuqtoplem 21309	. . . 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 682	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( a e. ( N ` p ) <-> E. w e. U a = ( w " { p } ) ) )
24	23	biimpa 491	. 2 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. w e. U a = ( w " { p } ) )
25	19, 24	r19.29a 2944	1 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> p e. a )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   E.wrex 2750   _Vcvv 3057   C_ wss 3416   {csn 3980   |-> cmpt 4477   _I cid 4766   ran crn 4857   |` cres 4858   "cima 4859   Fn wfn 5600   ` cfv 5605  UnifOncust 21269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ust 21270

This theorem is referenced by:  ustuqtop  21316  utopsnneiplem  21317