Theorem ballss3 39270

Description: A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
ballss3.y	|- F/ x ph
ballss3.d	|- ( ph -> D e. (PsMet ` X ) )
ballss3.p	|- ( ph -> P e. X )
ballss3.r	|- ( ph -> R e. RR* )
ballss3.a	|- ( ( ph /\ x e. X /\ ( P D x ) < R ) -> x e. A )

Assertion

Ref	Expression
ballss3	|- ( ph -> ( P ( ball ` D ) R ) C_ A )

Distinct variable groups:   x, A   x, D   x, P   x, R

Allowed substitution hints:   ph( x)   X( x)

Proof of Theorem ballss3

Step	Hyp	Ref	Expression
1		ballss3.y	. . 3 |- F/ x ph
2		simpl 473	. . . . 5 |- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ph )
3		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> x e. ( P ( ball ` D ) R ) )
4		ballss3.d	. . . . . . . . 9 |- ( ph -> D e. (PsMet ` X ) )
5		ballss3.p	. . . . . . . . 9 |- ( ph -> P e. X )
6		ballss3.r	. . . . . . . . 9 |- ( ph -> R e. RR* )
7		elblps 22192	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( x e. ( P ( ball ` D ) R ) <-> ( x e. X /\ ( P D x ) < R ) ) )
8	4, 5, 6, 7	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( x e. ( P ( ball ` D ) R ) <-> ( x e. X /\ ( P D x ) < R ) ) )
9	8	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( x e. ( P ( ball ` D ) R ) <-> ( x e. X /\ ( P D x ) < R ) ) )
10	3, 9	mpbid 222	. . . . . 6 |- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( x e. X /\ ( P D x ) < R ) )
11	10	simpld 475	. . . . 5 |- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> x e. X )
12	10	simprd 479	. . . . 5 |- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( P D x ) < R )
13		ballss3.a	. . . . 5 |- ( ( ph /\ x e. X /\ ( P D x ) < R ) -> x e. A )
14	2, 11, 12, 13	syl3anc 1326	. . . 4 |- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> x e. A )
15	14	ex 450	. . 3 |- ( ph -> ( x e. ( P ( ball ` D ) R ) -> x e. A ) )
16	1, 15	ralrimi 2957	. 2 |- ( ph -> A. x e. ( P ( ball ` D ) R ) x e. A )
17		dfss3 3592	. 2 |- ( ( P ( ball ` D ) R ) C_ A <-> A. x e. ( P ( ball ` D ) R ) x e. A )
18	16, 17	sylibr 224	1 |- ( ph -> ( P ( ball ` D ) R ) C_ A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   F/wnf 1708   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RR*cxr 10073   < clt 10074  PsMetcpsmet 19730   ballcbl 19733

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

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-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-pw 4160  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-xr 10078  df-psmet 19738  df-bl 19741

This theorem is referenced by:  ioorrnopnlem  40524