Theorem ustuqtop3 19816

Description: Lemma for ustuqtop 19819, similar to elnei 18713 (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 5526	. . . . . . 7 |- ( _I |` X ) Fn X
2		fnsnfv 5749	. . . . . . 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 19781	. . . . . . 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 5201	. . . . . 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 3388	. . . 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 5699	. . . . 5 |- ( ( _I |` X ) ` p ) e. _V
13	12	snss 3997	. . . 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 5902	. . . . 5 |- ( p e. X -> ( ( _I |` X ) ` p ) = p )
16	15	eqcomd 2446	. . . 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 2522	. 2 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) /\ w e. U ) /\ a = ( w " { p } ) ) -> p e. a )
20		vex 2973	. . . 4 |- a e. _V
21		utopustuq.1	. . . . 5 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
22	21	ustuqtoplem 19812	. . . 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 2860	1 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> p e. a )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   E.wrex 2714   _Vcvv 2970   C_ wss 3326   {csn 3875   e. cmpt 4348   _I cid 4629   ran crn 4839   |` cres 4840   "cima 4841   Fn wfn 5411   ` cfv 5416  UnifOncust 19772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ust 19773

This theorem is referenced by:  ustuqtop  19819  utopsnneiplem  19820