Theorem ustuqtop5 22049

Description: Lemma for ustuqtop 22050. (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
ustuqtop5	|- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> X e. ( N ` p ) )

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

Allowed substitution hint:   N( v)

Proof of Theorem ustuqtop5

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ustbasel 22010	. . . 4 |- ( U e. (UnifOn ` X ) -> ( X X. X ) e. U )
2	1	adantr 481	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( X X. X ) e. U )
3		snssi 4339	. . . . . . . . 9 |- ( p e. X -> { p } C_ X )
4		dfss 3589	. . . . . . . . 9 |- ( { p } C_ X <-> { p } = ( { p } i^i X ) )
5	3, 4	sylib 208	. . . . . . . 8 |- ( p e. X -> { p } = ( { p } i^i X ) )
6		incom 3805	. . . . . . . 8 |- ( { p } i^i X ) = ( X i^i { p } )
7	5, 6	syl6req 2673	. . . . . . 7 |- ( p e. X -> ( X i^i { p } ) = { p } )
8		snnzg 4308	. . . . . . 7 |- ( p e. X -> { p } =/= (/) )
9	7, 8	eqnetrd 2861	. . . . . 6 |- ( p e. X -> ( X i^i { p } ) =/= (/) )
10	9	adantl 482	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( X i^i { p } ) =/= (/) )
11		xpima2 5578	. . . . 5 |- ( ( X i^i { p } ) =/= (/) -> ( ( X X. X ) " { p } ) = X )
12	10, 11	syl 17	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( ( X X. X ) " { p } ) = X )
13	12	eqcomd 2628	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> X = ( ( X X. X ) " { p } ) )
14		imaeq1 5461	. . . . 5 |- ( w = ( X X. X ) -> ( w " { p } ) = ( ( X X. X ) " { p } ) )
15	14	eqeq2d 2632	. . . 4 |- ( w = ( X X. X ) -> ( X = ( w " { p } ) <-> X = ( ( X X. X ) " { p } ) ) )
16	15	rspcev 3309	. . 3 |- ( ( ( X X. X ) e. U /\ X = ( ( X X. X ) " { p } ) ) -> E. w e. U X = ( w " { p } ) )
17	2, 13, 16	syl2anc 693	. 2 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> E. w e. U X = ( w " { p } ) )
18		elfvex 6221	. . 3 |- ( U e. (UnifOn ` X ) -> X e. _V )
19		utopustuq.1	. . . 4 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
20	19	ustuqtoplem 22043	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ X e. _V ) -> ( X e. ( N ` p ) <-> E. w e. U X = ( w " { p } ) ) )
21	18, 20	mpidan 704	. 2 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( X e. ( N ` p ) <-> E. w e. U X = ( w " { p } ) ) )
22	17, 21	mpbird 247	1 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> X e. ( N ` p ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. cxp 5112   ran crn 5115   "cima 5117   ` 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