Theorem ustfn 22005

Description: The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)

Assertion

Ref	Expression
ustfn	|- UnifOn Fn _V

Proof of Theorem ustfn

Dummy variables v u w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		selpw 4165	. . . . 5 |- ( u e. ~P ~P ( x X. x ) <-> u C_ ~P ( x X. x ) )
2	1	abbii 2739	. . . 4 |- { u | u e. ~P ~P ( x X. x ) } = { u | u C_ ~P ( x X. x ) }
3		abid2 2745	. . . . 5 |- { u | u e. ~P ~P ( x X. x ) } = ~P ~P ( x X. x )
4		vex 3203	. . . . . . . 8 |- x e. _V
5	4, 4	xpex 6962	. . . . . . 7 |- ( x X. x ) e. _V
6	5	pwex 4848	. . . . . 6 |- ~P ( x X. x ) e. _V
7	6	pwex 4848	. . . . 5 |- ~P ~P ( x X. x ) e. _V
8	3, 7	eqeltri 2697	. . . 4 |- { u | u e. ~P ~P ( x X. x ) } e. _V
9	2, 8	eqeltrri 2698	. . 3 |- { u | u C_ ~P ( x X. x ) } e. _V
10		simp1 1061	. . . 4 |- ( ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) -> u C_ ~P ( x X. x ) )
11	10	ss2abi 3674	. . 3 |- { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } C_ { u | u C_ ~P ( x X. x ) }
12	9, 11	ssexi 4803	. 2 |- { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } e. _V
13		df-ust 22004	. 2 |- UnifOn = ( x e. _V |-> { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } )
14	12, 13	fnmpti 6022	1 |- UnifOn Fn _V

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   Fn wfn 5883  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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-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-fun 5890  df-fn 5891  df-ust 22004

This theorem is referenced by:  ustn0  22024  elrnust  22028  ustbas  22031