Theorem ustuqtop0 22044

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
ustuqtop0	|- ( U e. (UnifOn ` X ) -> N : X --> ~P ~P X )

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

Allowed substitution hint:   N( v)

Proof of Theorem ustuqtop0

Step	Hyp	Ref	Expression
1		ustimasn 22032	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ v e. U /\ p e. X ) -> ( v " { p } ) C_ X )
2	1	3expa 1265	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ v e. U ) /\ p e. X ) -> ( v " { p } ) C_ X )
3	2	an32s 846	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ v e. U ) -> ( v " { p } ) C_ X )
4		vex 3203	. . . . . . . 8 |- v e. _V
5	4	imaex 7104	. . . . . . 7 |- ( v " { p } ) e. _V
6	5	elpw 4164	. . . . . 6 |- ( ( v " { p } ) e. ~P X <-> ( v " { p } ) C_ X )
7	3, 6	sylibr 224	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ v e. U ) -> ( v " { p } ) e. ~P X )
8	7	ralrimiva 2966	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> A. v e. U ( v " { p } ) e. ~P X )
9		eqid 2622	. . . . 5 |- ( v e. U |-> ( v " { p } ) ) = ( v e. U |-> ( v " { p } ) )
10	9	rnmptss 6392	. . . 4 |- ( A. v e. U ( v " { p } ) e. ~P X -> ran ( v e. U |-> ( v " { p } ) ) C_ ~P X )
11	8, 10	syl 17	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ran ( v e. U |-> ( v " { p } ) ) C_ ~P X )
12		mptexg 6484	. . . . 5 |- ( U e. (UnifOn ` X ) -> ( v e. U |-> ( v " { p } ) ) e. _V )
13		rnexg 7098	. . . . 5 |- ( ( v e. U |-> ( v " { p } ) ) e. _V -> ran ( v e. U |-> ( v " { p } ) ) e. _V )
14		elpwg 4166	. . . . 5 |- ( ran ( v e. U |-> ( v " { p } ) ) e. _V -> ( ran ( v e. U |-> ( v " { p } ) ) e. ~P ~P X <-> ran ( v e. U |-> ( v " { p } ) ) C_ ~P X ) )
15	12, 13, 14	3syl 18	. . . 4 |- ( U e. (UnifOn ` X ) -> ( ran ( v e. U |-> ( v " { p } ) ) e. ~P ~P X <-> ran ( v e. U |-> ( v " { p } ) ) C_ ~P X ) )
16	15	adantr 481	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( ran ( v e. U |-> ( v " { p } ) ) e. ~P ~P X <-> ran ( v e. U |-> ( v " { p } ) ) C_ ~P X ) )
17	11, 16	mpbird 247	. 2 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ran ( v e. U |-> ( v " { p } ) ) e. ~P ~P X )
18		utopustuq.1	. 2 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
19	17, 18	fmptd 6385	1 |- ( U e. (UnifOn ` X ) -> N : X --> ~P ~P X )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   ran crn 5115   "cima 5117   -->wf 5884   ` 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