Theorem ustuqtoplem 21265

Description: Lemma for ustuqtop 21272. (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 463	. . . . . . . . . . 11 |- ( ( p = q /\ v e. U ) -> p = q )
3	2	sneqd 3948	. . . . . . . . . 10 |- ( ( p = q /\ v e. U ) -> { p } = { q } )
4	3	imaeq2d 5146	. . . . . . . . 9 |- ( ( p = q /\ v e. U ) -> ( v " { p } ) = ( v " { q } ) )
5	4	mpteq2dva 4461	. . . . . . . 8 |- ( p = q -> ( v e. U |-> ( v " { p } ) ) = ( v e. U |-> ( v " { q } ) ) )
6	5	rneqd 5040	. . . . . . 7 |- ( p = q -> ran ( v e. U |-> ( v " { p } ) ) = ran ( v e. U |-> ( v " { q } ) ) )
7	6	cbvmptv 4467	. . . . . 6 |- ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) ) = ( q e. X |-> ran ( v e. U |-> ( v " { q } ) ) )
8	1, 7	eqtri 2474	. . . . 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 1016	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> q = P )
11	10	sneqd 3948	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> { q } = { P } )
12	11	imaeq2d 5146	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ ( P e. X /\ q = P /\ v e. U ) ) -> ( v " { q } ) = ( v " { P } ) )
13	12	3anassrs 1235	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) /\ v e. U ) -> ( v " { q } ) = ( v " { P } ) )
14	13	mpteq2dva 4461	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) -> ( v e. U |-> ( v " { q } ) ) = ( v e. U |-> ( v " { P } ) ) )
15	14	rneqd 5040	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ P e. X ) /\ q = P ) -> ran ( v e. U |-> ( v " { q } ) ) = ran ( v e. U |-> ( v " { P } ) ) )
16		simpr 467	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> P e. X )
17		mptexg 6121	. . . . . 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 17	. . . . 5 |- ( U e. (UnifOn ` X ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
20	19	adantr 471	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ran ( v e. U |-> ( v " { P } ) ) e. _V )
21	9, 15, 16, 20	fvmptd 5938	. . 3 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( N ` P ) = ran ( v e. U |-> ( v " { P } ) ) )
22	21	eleq2d 2515	. 2 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( A e. ( N ` P ) <-> A e. ran ( v e. U |-> ( v " { P } ) ) ) )
23		imaeq1 5141	. . . 4 |- ( v = w -> ( v " { P } ) = ( w " { P } ) )
24	23	cbvmptv 4467	. . 3 |- ( v e. U |-> ( v " { P } ) ) = ( w e. U |-> ( w " { P } ) )
25	24	elrnmpt 5059	. 2 |- ( A e. V -> ( A e. ran ( v e. U |-> ( v " { P } ) ) <-> E. w e. U A = ( w " { P } ) ) )
26	22, 25	sylan9bb 711	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 189   /\ wa 375   /\ w3a 986   = wceq 1448   e. wcel 1891   E.wrex 2738   _Vcvv 3013   {csn 3936   |-> cmpt 4433   ran crn 4813   "cima 4815   ` cfv 5561  UnifOncust 21225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pr 4612  ax-un 6571

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569

This theorem is referenced by:  ustuqtop1  21267  ustuqtop2  21268  ustuqtop3  21269  ustuqtop4  21270  ustuqtop5  21271  utopsnneiplem  21273