Theorem pjf 20057

Description: A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)

Hypotheses

Ref	Expression
pjf.k	|- K = ( proj ` W )
pjf.v	|- V = ( Base ` W )

Assertion

Ref	Expression
pjf	|- ( T e. dom K -> ( K ` T ) : V --> V )

Proof of Theorem pjf

Step	Hyp	Ref	Expression
1		pjf.v	. . . 4 |- V = ( Base ` W )
2		eqid 2622	. . . 4 |- ( LSubSp ` W ) = ( LSubSp ` W )
3		eqid 2622	. . . 4 |- ( ocv ` W ) = ( ocv ` W )
4		eqid 2622	. . . 4 |- ( proj1 ` W ) = ( proj1 ` W )
5		pjf.k	. . . 4 |- K = ( proj ` W )
6	1, 2, 3, 4, 5	pjdm 20051	. . 3 |- ( T e. dom K <-> ( T e. ( LSubSp ` W ) /\ ( T ( proj1 ` W ) ( ( ocv ` W ) ` T ) ) : V --> V ) )
7	6	simprbi 480	. 2 |- ( T e. dom K -> ( T ( proj1 ` W ) ( ( ocv ` W ) ` T ) ) : V --> V )
8	3, 4, 5	pjval 20054	. . 3 |- ( T e. dom K -> ( K ` T ) = ( T ( proj1 ` W ) ( ( ocv ` W ) ` T ) ) )
9	8	feq1d 6030	. 2 |- ( T e. dom K -> ( ( K ` T ) : V --> V <-> ( T ( proj1 ` W ) ( ( ocv ` W ) ` T ) ) : V --> V ) )
10	7, 9	mpbird 247	1 |- ( T e. dom K -> ( K ` T ) : V --> V )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   proj1cpj1 18050   LSubSpclss 18932   ocvcocv 20004   projcpj 20044

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

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-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-map 7859  df-pj 20047

This theorem is referenced by: (None)