Theorem idlaut 36236

Description: The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)

Hypotheses

Ref	Expression
idlaut.b	|- B = ( Base ` K )
idlaut.i	|- I = ( LAut ` K )

Assertion

Ref	Expression
idlaut	|- ( K e. A -> ( _I |` B ) e. I )

Proof of Theorem idlaut

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 5833	. . 3 |- ( _I |` B ) : B -1-1-onto-> B
2	1	a1i 11	. 2 |- ( K e. A -> ( _I |` B ) : B -1-1-onto-> B )
3		fvresi 6073	. . . . . 6 |- ( x e. B -> ( ( _I |` B ) ` x ) = x )
4		fvresi 6073	. . . . . 6 |- ( y e. B -> ( ( _I |` B ) ` y ) = y )
5	3, 4	breqan12d 4454	. . . . 5 |- ( ( x e. B /\ y e. B ) -> ( ( ( _I |` B ) ` x ) ( le ` K ) ( ( _I |` B ) ` y ) <-> x ( le ` K ) y ) )
6	5	bicomd 201	. . . 4 |- ( ( x e. B /\ y e. B ) -> ( x ( le ` K ) y <-> ( ( _I |` B ) ` x ) ( le ` K ) ( ( _I |` B ) ` y ) ) )
7	6	rgen2a 2881	. . 3 |- A. x e. B A. y e. B ( x ( le ` K ) y <-> ( ( _I |` B ) ` x ) ( le ` K ) ( ( _I |` B ) ` y ) )
8	7	a1i 11	. 2 |- ( K e. A -> A. x e. B A. y e. B ( x ( le ` K ) y <-> ( ( _I |` B ) ` x ) ( le ` K ) ( ( _I |` B ) ` y ) ) )
9		idlaut.b	. . 3 |- B = ( Base ` K )
10		eqid 2454	. . 3 |- ( le ` K ) = ( le ` K )
11		idlaut.i	. . 3 |- I = ( LAut ` K )
12	9, 10, 11	islaut 36223	. 2 |- ( K e. A -> ( ( _I |` B ) e. I <-> ( ( _I |` B ) : B -1-1-onto-> B /\ A. x e. B A. y e. B ( x ( le ` K ) y <-> ( ( _I |` B ) ` x ) ( le ` K ) ( ( _I |` B ) ` y ) ) ) ) )
13	2, 8, 12	mpbir2and 920	1 |- ( K e. A -> ( _I |` B ) e. I )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   class class class wbr 4439   _I cid 4779   |` cres 4990   -1-1-onto->wf1o 5569   ` cfv 5570   Basecbs 14719   lecple 14794   LAutclaut 36125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-laut 36129

This theorem is referenced by:  idldil  36254