Theorem ustuqtoplem 20477

Description: Lemma for ustuqtop 20484 (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 457	. . . . . . . . . . 11 |- ( ( p = q /\ v e. U ) -> p = q )
3	2	sneqd 4039	. . . . . . . . . 10 |- ( ( p = q /\ v e. U ) -> { p } = { q } )
4	3	imaeq2d 5335	. . . . . . . . 9 |- ( ( p = q /\ v e. U ) -> ( v " { p } ) = ( v " { q } ) )
5	4	mpteq2dva 4533	. . . . . . . 8 |- ( p = q -> ( v e. U |-> ( v " { p } ) ) = ( v e. U |-> ( v " { q } ) ) )
6	5	rneqd 5228	. . . . . . 7 |- ( p = q -> ran ( v e. U |-> ( v " { p } ) ) = ran ( v e. U |-> ( v " { q } ) ) )
7	6	cbvmptv 4538	. . . . . 6 |- ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) ) = ( q e. X |-> ran ( v e. U |-> ( v " { q } ) ) )
8	1, 7	eqtri 2496	. . . . 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 1003	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> q = P )
11	10	sneqd 4039	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> { q } = { P } )
12	11	imaeq2d 5335	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> ( v " { q } ) = ( v " { P } ) )
13	12	3anassrs 1218	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) /\ v e. U ) -> ( v " { q } ) = ( v " { P } ) )
14	13	mpteq2dva 4533	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) -> ( v e. U |-> ( v " { q } ) ) = ( v e. U |-> ( v " { P } ) ) )
15	14	rneqd 5228	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) -> ran ( v e. U |-> ( v " { q } ) ) = ran ( v e. U |-> ( v " { P } ) ) )
16		simpr 461	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> P e. X )
17		mptexg 6128	. . . . . 6 |- ( U e. (UnifOn ` X ) -> ( v e. U |-> ( v " { P } ) ) e. _V )
18		rnexg 6713	. . . . . 6 |- ( ( v e. U |-> ( v " { P } ) ) e. _V -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
19	17, 18	syl 16	. . . . 5 |- ( U e. (UnifOn ` X ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
20	19	adantr 465	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
21	9, 15, 16, 20	fvmptd 5953	. . 3 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( N ` P ) = ran ( v e. U |-> ( v " { P } ) ) )
22	21	eleq2d 2537	. 2 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( A e. ( N ` P ) <-> A e. ran ( v e. U |-> ( v " { P } ) ) ) )
23		imaeq1 5330	. . . 4 |- ( v = w -> ( v " { P } ) = ( w " { P } ) )
24	23	cbvmptv 4538	. . 3 |- ( v e. U |-> ( v " { P } ) ) = ( w e. U |-> ( w " { P } ) )
25	24	elrnmpt 5247	. 2 |- ( A e. V -> ( A e. ran ( v e. U |-> ( v " { P } ) ) <-> E. w e. U A = ( w " { P } ) ) )
26	22, 25	sylan9bb 699	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 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   E.wrex 2815   _Vcvv 3113   {csn 4027   |-> cmpt 4505   ran crn 5000   "cima 5002   ` cfv 5586  UnifOncust 20437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594

This theorem is referenced by:  ustuqtop1  20479  ustuqtop2  20480  ustuqtop3  20481  ustuqtop4  20482  ustuqtop5  20483  utopsnneiplem  20485