Theorem ustbas2 22029

Description: Second direction for ustbas 22031. (Contributed by Thierry Arnoux, 16-Nov-2017.)

Assertion

Ref	Expression
ustbas2	|- ( U e. (UnifOn ` X ) -> X = dom U. U )

Proof of Theorem ustbas2

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

Step	Hyp	Ref	Expression
1		dmxpid 5345	. 2 |- dom ( X X. X ) = X
2		ustbasel 22010	. . . . 5 |- ( U e. (UnifOn ` X ) -> ( X X. X ) e. U )
3		elssuni 4467	. . . . 5 |- ( ( X X. X ) e. U -> ( X X. X ) C_ U. U )
4	2, 3	syl 17	. . . 4 |- ( U e. (UnifOn ` X ) -> ( X X. X ) C_ U. U )
5		elfvex 6221	. . . . . . . . 9 |- ( U e. (UnifOn ` X ) -> X e. _V )
6		isust 22007	. . . . . . . . 9 |- ( X e. _V -> ( U e. (UnifOn ` X ) <-> ( 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 ) ) ) ) )
7	5, 6	syl 17	. . . . . . . 8 |- ( U e. (UnifOn ` X ) -> ( U e. (UnifOn ` X ) <-> ( 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 ) ) ) ) )
8	7	ibi 256	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> ( 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 ) ) ) )
9	8	simp1d 1073	. . . . . 6 |- ( U e. (UnifOn ` X ) -> U C_ ~P ( X X. X ) )
10	9	unissd 4462	. . . . 5 |- ( U e. (UnifOn ` X ) -> U. U C_ U. ~P ( X X. X ) )
11		unipw 4918	. . . . 5 |- U. ~P ( X X. X ) = ( X X. X )
12	10, 11	syl6sseq 3651	. . . 4 |- ( U e. (UnifOn ` X ) -> U. U C_ ( X X. X ) )
13	4, 12	eqssd 3620	. . 3 |- ( U e. (UnifOn ` X ) -> ( X X. X ) = U. U )
14	13	dmeqd 5326	. 2 |- ( U e. (UnifOn ` X ) -> dom ( X X. X ) = dom U. U )
15	1, 14	syl5eqr 2670	1 |- ( U e. (UnifOn ` X ) -> X = dom U. U )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   |` cres 5116   o. ccom 5118   ` 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-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-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-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-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ust 22004

This theorem is referenced by:  ustbas  22031  utopval  22036  tuslem  22071  ucnval  22081  iscfilu  22092