Theorem ustuqtoplem 21252

Description: Lemma for ustuqtop 21259 (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 458	. . . . . . . . . . 11 |- ( ( p = q /\ v e. U ) -> p = q )
3	2	sneqd 4010	. . . . . . . . . 10 |- ( ( p = q /\ v e. U ) -> { p } = { q } )
4	3	imaeq2d 5187	. . . . . . . . 9 |- ( ( p = q /\ v e. U ) -> ( v " { p } ) = ( v " { q } ) )
5	4	mpteq2dva 4510	. . . . . . . 8 |- ( p = q -> ( v e. U |-> ( v " { p } ) ) = ( v e. U |-> ( v " { q } ) ) )
6	5	rneqd 5081	. . . . . . 7 |- ( p = q -> ran ( v e. U |-> ( v " { p } ) ) = ran ( v e. U |-> ( v " { q } ) ) )
7	6	cbvmptv 4516	. . . . . 6 |- ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) ) = ( q e. X |-> ran ( v e. U |-> ( v " { q } ) ) )
8	1, 7	eqtri 2451	. . . . 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 1012	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> q = P )
11	10	sneqd 4010	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> { q } = { P } )
12	11	imaeq2d 5187	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> ( v " { q } ) = ( v " { P } ) )
13	12	3anassrs 1228	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) /\ v e. U ) -> ( v " { q } ) = ( v " { P } ) )
14	13	mpteq2dva 4510	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) -> ( v e. U |-> ( v " { q } ) ) = ( v e. U |-> ( v " { P } ) ) )
15	14	rneqd 5081	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) -> ran ( v e. U |-> ( v " { q } ) ) = ran ( v e. U |-> ( v " { P } ) ) )
16		simpr 462	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> P e. X )
17		mptexg 6150	. . . . . 6 |- ( U e. (UnifOn ` X ) -> ( v e. U |-> ( v " { P } ) ) e. _V )
18		rnexg 6739	. . . . . 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 466	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
21	9, 15, 16, 20	fvmptd 5970	. . 3 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( N ` P ) = ran ( v e. U |-> ( v " { P } ) ) )
22	21	eleq2d 2492	. 2 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( A e. ( N ` P ) <-> A e. ran ( v e. U |-> ( v " { P } ) ) ) )
23		imaeq1 5182	. . . 4 |- ( v = w -> ( v " { P } ) = ( w " { P } ) )
24	23	cbvmptv 4516	. . 3 |- ( v e. U |-> ( v " { P } ) ) = ( w e. U |-> ( w " { P } ) )
25	24	elrnmpt 5100	. 2 |- ( A e. V -> ( A e. ran ( v e. U |-> ( v " { P } ) ) <-> E. w e. U A = ( w " { P } ) ) )
26	22, 25	sylan9bb 704	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 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   E.wrex 2772   _Vcvv 3080   {csn 3998   |-> cmpt 4482   ran crn 4854   "cima 4856   ` cfv 5601  UnifOncust 21212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609

This theorem is referenced by:  ustuqtop1  21254  ustuqtop2  21255  ustuqtop3  21256  ustuqtop4  21257  ustuqtop5  21258  utopsnneiplem  21260