Theorem clsk1indlem0 38339

Description: The ansatz closure function ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) ) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.)

Hypothesis

Ref	Expression
clsk1indlem.k	|- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )

Assertion

Ref	Expression
clsk1indlem0	|- ( K ` (/) ) = (/)

Proof of Theorem clsk1indlem0

Step	Hyp	Ref	Expression
1		0elpw 4834	. 2 |- (/) e. ~P 3o
2		eqeq1 2626	. . . . 5 |- ( r = (/) -> ( r = { (/) } <-> (/) = { (/) } ) )
3		id 22	. . . . 5 |- ( r = (/) -> r = (/) )
4	2, 3	ifbieq2d 4111	. . . 4 |- ( r = (/) -> if ( r = { (/) } , { (/) , 1o } , r ) = if ( (/) = { (/) } , { (/) , 1o } , (/) ) )
5		0nep0 4836	. . . . . . 7 |- (/) =/= { (/) }
6	5	a1i 11	. . . . . 6 |- ( r = (/) -> (/) =/= { (/) } )
7	6	neneqd 2799	. . . . 5 |- ( r = (/) -> -. (/) = { (/) } )
8	7	iffalsed 4097	. . . 4 |- ( r = (/) -> if ( (/) = { (/) } , { (/) , 1o } , (/) ) = (/) )
9	4, 8	eqtrd 2656	. . 3 |- ( r = (/) -> if ( r = { (/) } , { (/) , 1o } , r ) = (/) )
10		clsk1indlem.k	. . 3 |- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )
11		0ex 4790	. . 3 |- (/) e. _V
12	9, 10, 11	fvmpt 6282	. 2 |- ( (/) e. ~P 3o -> ( K ` (/) ) = (/) )
13	1, 12	ax-mp 5	1 |- ( K ` (/) ) = (/)

Syntax hints:   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   {cpr 4179   |-> cmpt 4729   ` cfv 5888   1oc1o 7553   3oc3o 7555

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-ne 2795  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-pw 4160  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

This theorem is referenced by:  clsk1independent  38344