Theorem ustuqtoplem 19814

Description: Lemma for ustuqtop 19821 (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 3889	. . . . . . . . . 10 |- ( ( p = q /\ v e. U ) -> { p } = { q } )
4	3	imaeq2d 5169	. . . . . . . . 9 |- ( ( p = q /\ v e. U ) -> ( v " { p } ) = ( v " { q } ) )
5	4	mpteq2dva 4378	. . . . . . . 8 |- ( p = q -> ( v e. U |-> ( v " { p } ) ) = ( v e. U |-> ( v " { q } ) ) )
6	5	rneqd 5067	. . . . . . 7 |- ( p = q -> ran ( v e. U |-> ( v " { p } ) ) = ran ( v e. U |-> ( v " { q } ) ) )
7	6	cbvmptv 4383	. . . . . 6 |- ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) ) = ( q e. X |-> ran ( v e. U |-> ( v " { q } ) ) )
8	1, 7	eqtri 2463	. . . . 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 995	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> q = P )
11	10	sneqd 3889	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> { q } = { P } )
12	11	imaeq2d 5169	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> ( v " { q } ) = ( v " { P } ) )
13	12	3anassrs 1209	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) /\ v e. U ) -> ( v " { q } ) = ( v " { P } ) )
14	13	mpteq2dva 4378	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) -> ( v e. U |-> ( v " { q } ) ) = ( v e. U |-> ( v " { P } ) ) )
15	14	rneqd 5067	. . . 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 5947	. . . . . 6 |- ( U e. (UnifOn ` X ) -> ( v e. U |-> ( v " { P } ) ) e. _V )
18		rnexg 6510	. . . . . 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 5779	. . 3 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( N ` P ) = ran ( v e. U |-> ( v " { P } ) ) )
22	21	eleq2d 2510	. 2 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( A e. ( N ` P ) <-> A e. ran ( v e. U |-> ( v " { P } ) ) ) )
23		imaeq1 5164	. . . 4 |- ( v = w -> ( v " { P } ) = ( w " { P } ) )
24	23	cbvmptv 4383	. . 3 |- ( v e. U |-> ( v " { P } ) ) = ( w e. U |-> ( w " { P } ) )
25	24	elrnmpt 5086	. 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 965   = wceq 1369   e. wcel 1756   E.wrex 2716   _Vcvv 2972   {csn 3877   e. cmpt 4350   ran crn 4841   "cima 4843   ` cfv 5418  UnifOncust 19774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426

This theorem is referenced by:  ustuqtop1  19816  ustuqtop2  19817  ustuqtop3  19818  ustuqtop4  19819  ustuqtop5  19820  utopsnneiplem  19822