Theorem elhoi 40756

Description: Membership in a multidimensional half-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypothesis

Ref	Expression
elhoi.1	|- ( ph -> X e. V )

Assertion

Ref	Expression
elhoi	|- ( ph -> ( Y e. ( ( A [,) B ) ^m X ) <-> ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) ) )

Distinct variable groups:   x, A   x, B   x, X   x, Y

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

Proof of Theorem elhoi

Step	Hyp	Ref	Expression
1		ovexd 6680	. . 3 |- ( ph -> ( A [,) B ) e. _V )
2		elhoi.1	. . 3 |- ( ph -> X e. V )
3		elmapg 7870	. . 3 |- ( ( ( A [,) B ) e. _V /\ X e. V ) -> ( Y e. ( ( A [,) B ) ^m X ) <-> Y : X --> ( A [,) B ) ) )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> ( Y e. ( ( A [,) B ) ^m X ) <-> Y : X --> ( A [,) B ) ) )
5		id 22	. . . . . 6 |- ( Y : X --> ( A [,) B ) -> Y : X --> ( A [,) B ) )
6		icossxr 12258	. . . . . . 7 |- ( A [,) B ) C_ RR*
7	6	a1i 11	. . . . . 6 |- ( Y : X --> ( A [,) B ) -> ( A [,) B ) C_ RR* )
8	5, 7	fssd 6057	. . . . 5 |- ( Y : X --> ( A [,) B ) -> Y : X --> RR* )
9		ffvelrn 6357	. . . . . 6 |- ( ( Y : X --> ( A [,) B ) /\ x e. X ) -> ( Y ` x ) e. ( A [,) B ) )
10	9	ralrimiva 2966	. . . . 5 |- ( Y : X --> ( A [,) B ) -> A. x e. X ( Y ` x ) e. ( A [,) B ) )
11	8, 10	jca 554	. . . 4 |- ( Y : X --> ( A [,) B ) -> ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) )
12		ffn 6045	. . . . . . 7 |- ( Y : X --> RR* -> Y Fn X )
13	12	adantr 481	. . . . . 6 |- ( ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) -> Y Fn X )
14		simpr 477	. . . . . 6 |- ( ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) -> A. x e. X ( Y ` x ) e. ( A [,) B ) )
15	13, 14	jca 554	. . . . 5 |- ( ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) -> ( Y Fn X /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) )
16		ffnfv 6388	. . . . 5 |- ( Y : X --> ( A [,) B ) <-> ( Y Fn X /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) )
17	15, 16	sylibr 224	. . . 4 |- ( ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) -> Y : X --> ( A [,) B ) )
18	11, 17	impbii 199	. . 3 |- ( Y : X --> ( A [,) B ) <-> ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) )
19	18	a1i 11	. 2 |- ( ph -> ( Y : X --> ( A [,) B ) <-> ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) ) )
20	4, 19	bitrd 268	1 |- ( ph -> ( Y e. ( ( A [,) B ) ^m X ) <-> ( Y : X --> RR* /\ A. x e. X ( Y ` x ) e. ( A [,) B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   RR*cxr 10073   [,)cico 12177

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-ico 12181

This theorem is referenced by: (None)