Theorem ioorval 23342

Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)

Hypothesis

Ref	Expression
ioorf.1	|- F = ( x e. ran (,) |-> if ( x = (/) , <. 0 , 0 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) )

Assertion

Ref	Expression
ioorval	|- ( A e. ran (,) -> ( F ` A ) = if ( A = (/) , <. 0 , 0 >. , <.inf ( A , RR* , < ) , sup ( A , RR* , < ) >. ) )

Distinct variable group:   x, A

Allowed substitution hint:   F( x)

Proof of Theorem ioorval

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . 3 |- ( x = A -> ( x = (/) <-> A = (/) ) )
2		infeq1 8382	. . . 4 |- ( x = A -> inf ( x , RR* , < ) = inf ( A , RR* , < ) )
3		supeq1 8351	. . . 4 |- ( x = A -> sup ( x , RR* , < ) = sup ( A , RR* , < ) )
4	2, 3	opeq12d 4410	. . 3 |- ( x = A -> <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. = <.inf ( A , RR* , < ) , sup ( A , RR* , < ) >. )
5	1, 4	ifbieq2d 4111	. 2 |- ( x = A -> if ( x = (/) , <. 0 , 0 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) = if ( A = (/) , <. 0 , 0 >. , <.inf ( A , RR* , < ) , sup ( A , RR* , < ) >. ) )
6		ioorf.1	. 2 |- F = ( x e. ran (,) |-> if ( x = (/) , <. 0 , 0 >. , <.inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) )
7		opex 4932	. . 3 |- <. 0 , 0 >. e. _V
8		opex 4932	. . 3 |- <.inf ( A , RR* , < ) , sup ( A , RR* , < ) >. e. _V
9	7, 8	ifex 4156	. 2 |- if ( A = (/) , <. 0 , 0 >. , <.inf ( A , RR* , < ) , sup ( A , RR* , < ) >. ) e. _V
10	5, 6, 9	fvmpt 6282	1 |- ( A e. ran (,) -> ( F ` A ) = if ( A = (/) , <. 0 , 0 >. , <.inf ( A , RR* , < ) , sup ( A , RR* , < ) >. ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   (/)c0 3915   ifcif 4086   <.cop 4183   |-> cmpt 4729   ran crn 5115   ` cfv 5888   supcsup 8346  infcinf 8347   0cc0 9936   RR*cxr 10073   < clt 10074   (,)cioo 12175

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-iota 5851  df-fun 5890  df-fv 5896  df-sup 8348  df-inf 8349

This theorem is referenced by:  ioorinv2  23343  ioorinv  23344  ioorcl  23345