Theorem ustuqtoplem 22043

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
ustuqtoplem	|- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ A e. V ) -> ( A e. ( N ` P ) <-> E. w e. U A = ( w " { P } ) ) )

Distinct variable groups:   w, A   w, v, P   v, p, w, U   X, p, v

Allowed substitution hints:   A( v, p)   P( p)   N( w, v, p)   V( w, v, p)   X( w)

Proof of Theorem ustuqtoplem

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		utopustuq.1	. . . . . 6 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
2		simpl 473	. . . . . . . . . . 11 |- ( ( p = q /\ v e. U ) -> p = q )
3	2	sneqd 4189	. . . . . . . . . 10 |- ( ( p = q /\ v e. U ) -> { p } = { q } )
4	3	imaeq2d 5466	. . . . . . . . 9 |- ( ( p = q /\ v e. U ) -> ( v " { p } ) = ( v " { q } ) )
5	4	mpteq2dva 4744	. . . . . . . 8 |- ( p = q -> ( v e. U |-> ( v " { p } ) ) = ( v e. U |-> ( v " { q } ) ) )
6	5	rneqd 5353	. . . . . . 7 |- ( p = q -> ran ( v e. U |-> ( v " { p } ) ) = ran ( v e. U |-> ( v " { q } ) ) )
7	6	cbvmptv 4750	. . . . . 6 |- ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) ) = ( q e. X |-> ran ( v e. U |-> ( v " { q } ) ) )
8	1, 7	eqtri 2644	. . . . 5 |- N = ( q e. X |-> ran ( v e. U |-> ( v " { q } ) ) )
9	8	a1i 11	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> N = ( q e. X |-> ran ( v e. U |-> ( v " { q } ) ) ) )
10		simpr2 1068	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> q = P )
11	10	sneqd 4189	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> { q } = { P } )
12	11	imaeq2d 5466	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> ( v " { q } ) = ( v " { P } ) )
13	12	3anassrs 1290	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) /\ v e. U ) -> ( v " { q } ) = ( v " { P } ) )
14	13	mpteq2dva 4744	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) -> ( v e. U |-> ( v " { q } ) ) = ( v e. U |-> ( v " { P } ) ) )
15	14	rneqd 5353	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) -> ran ( v e. U |-> ( v " { q } ) ) = ran ( v e. U |-> ( v " { P } ) ) )
16		simpr 477	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> P e. X )
17		mptexg 6484	. . . . . 6 |- ( U e. (UnifOn ` X ) -> ( v e. U |-> ( v " { P } ) ) e. _V )
18		rnexg 7098	. . . . . 6 |- ( ( v e. U |-> ( v " { P } ) ) e. _V -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
19	17, 18	syl 17	. . . . 5 |- ( U e. (UnifOn ` X ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
20	19	adantr 481	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
21	9, 15, 16, 20	fvmptd 6288	. . 3 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( N ` P ) = ran ( v e. U |-> ( v " { P } ) ) )
22	21	eleq2d 2687	. 2 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( A e. ( N ` P ) <-> A e. ran ( v e. U |-> ( v " { P } ) ) ) )
23		imaeq1 5461	. . . 4 |- ( v = w -> ( v " { P } ) = ( w " { P } ) )
24	23	cbvmptv 4750	. . 3 |- ( v e. U |-> ( v " { P } ) ) = ( w e. U |-> ( w " { P } ) )
25	24	elrnmpt 5372	. 2 |- ( A e. V -> ( A e. ran ( v e. U |-> ( v " { P } ) ) <-> E. w e. U A = ( w " { P } ) ) )
26	22, 25	sylan9bb 736	1 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ A e. V ) -> ( A e. ( N ` P ) <-> E. w e. U A = ( w " { P } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   {csn 4177   |-> cmpt 4729   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-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-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

This theorem is referenced by:  ustuqtop1  22045  ustuqtop2  22046  ustuqtop3  22047  ustuqtop4  22048  ustuqtop5  22049  utopsnneiplem  22051