Theorem hoi2toco 40821

Description: The half-open interval expressed using a composition of a function into ( RR X. RR ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
hoi2toco.1	|- F/ k ph
hoi2toco.c	|- I = ( k e. X |-> <. ( A ` k ) , ( B ` k ) >. )

Assertion

Ref	Expression
hoi2toco	|- ( ph -> X_ k e. X ( ( [,) o. I ) ` k ) = X_ k e. X ( ( A ` k ) [,) ( B ` k ) ) )

Distinct variable group:   k, X

Allowed substitution hints:   ph( k)   A( k)   B( k)   I( k)

Proof of Theorem hoi2toco

Step	Hyp	Ref	Expression
1		hoi2toco.1	. 2 |- F/ k ph
2		hoi2toco.c	. . . . . . 7 |- I = ( k e. X |-> <. ( A ` k ) , ( B ` k ) >. )
3	2	funmpt2 5927	. . . . . 6 |- Fun I
4	3	a1i 11	. . . . 5 |- ( ph -> Fun I )
5	4	adantr 481	. . . 4 |- ( ( ph /\ k e. X ) -> Fun I )
6		simpr 477	. . . . 5 |- ( ( ph /\ k e. X ) -> k e. X )
7	2	dmeqi 5325	. . . . . . . 8 |- dom I = dom ( k e. X |-> <. ( A ` k ) , ( B ` k ) >. )
8	7	a1i 11	. . . . . . 7 |- ( ph -> dom I = dom ( k e. X |-> <. ( A ` k ) , ( B ` k ) >. ) )
9		opex 4932	. . . . . . . . . 10 |- <. ( A ` k ) , ( B ` k ) >. e. _V
10	9	2a1i 12	. . . . . . . . 9 |- ( ph -> ( k e. X -> <. ( A ` k ) , ( B ` k ) >. e. _V ) )
11	1, 10	ralrimi 2957	. . . . . . . 8 |- ( ph -> A. k e. X <. ( A ` k ) , ( B ` k ) >. e. _V )
12		dmmptg 5632	. . . . . . . 8 |- ( A. k e. X <. ( A ` k ) , ( B ` k ) >. e. _V -> dom ( k e. X |-> <. ( A ` k ) , ( B ` k ) >. ) = X )
13	11, 12	syl 17	. . . . . . 7 |- ( ph -> dom ( k e. X |-> <. ( A ` k ) , ( B ` k ) >. ) = X )
14	8, 13	eqtr2d 2657	. . . . . 6 |- ( ph -> X = dom I )
15	14	adantr 481	. . . . 5 |- ( ( ph /\ k e. X ) -> X = dom I )
16	6, 15	eleqtrd 2703	. . . 4 |- ( ( ph /\ k e. X ) -> k e. dom I )
17		fvco 6274	. . . 4 |- ( ( Fun I /\ k e. dom I ) -> ( ( [,) o. I ) ` k ) = ( [,) ` ( I ` k ) ) )
18	5, 16, 17	syl2anc 693	. . 3 |- ( ( ph /\ k e. X ) -> ( ( [,) o. I ) ` k ) = ( [,) ` ( I ` k ) ) )
19	9	a1i 11	. . . . 5 |- ( ( ph /\ k e. X ) -> <. ( A ` k ) , ( B ` k ) >. e. _V )
20	2	fvmpt2 6291	. . . . 5 |- ( ( k e. X /\ <. ( A ` k ) , ( B ` k ) >. e. _V ) -> ( I ` k ) = <. ( A ` k ) , ( B ` k ) >. )
21	6, 19, 20	syl2anc 693	. . . 4 |- ( ( ph /\ k e. X ) -> ( I ` k ) = <. ( A ` k ) , ( B ` k ) >. )
22	21	fveq2d 6195	. . 3 |- ( ( ph /\ k e. X ) -> ( [,) ` ( I ` k ) ) = ( [,) ` <. ( A ` k ) , ( B ` k ) >. ) )
23		df-ov 6653	. . . . 5 |- ( ( A ` k ) [,) ( B ` k ) ) = ( [,) ` <. ( A ` k ) , ( B ` k ) >. )
24	23	eqcomi 2631	. . . 4 |- ( [,) ` <. ( A ` k ) , ( B ` k ) >. ) = ( ( A ` k ) [,) ( B ` k ) )
25	24	a1i 11	. . 3 |- ( ( ph /\ k e. X ) -> ( [,) ` <. ( A ` k ) , ( B ` k ) >. ) = ( ( A ` k ) [,) ( B ` k ) ) )
26	18, 22, 25	3eqtrd 2660	. 2 |- ( ( ph /\ k e. X ) -> ( ( [,) o. I ) ` k ) = ( ( A ` k ) [,) ( B ` k ) ) )
27	1, 26	ixpeq2d 39237	1 |- ( ph -> X_ k e. X ( ( [,) o. I ) ` k ) = X_ k e. X ( ( A ` k ) [,) ( B ` k ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   X_cixp 7908   [,)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

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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-fv 5896  df-ov 6653  df-ixp 7909

This theorem is referenced by:  opnvonmbllem1  40846