Theorem pjfni 26281

Description: Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
pjfn.1	|- H e. CH

Assertion

Ref	Expression
pjfni	|- ( proj h ` H ) Fn ~H

Proof of Theorem pjfni

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

Step	Hyp	Ref	Expression
1		riotaex 6240	. 2 |- ( iota_ y e. H E. z e. ( _|_ ` H ) x = ( y +h z ) ) e. _V
2		pjfn.1	. . 3 |- H e. CH
3		pjhfval 25976	. . 3 |- ( H e. CH -> ( proj h ` H ) = ( x e. ~H |-> ( iota_ y e. H E. z e. ( _|_ ` H ) x = ( y +h z ) ) ) )
4	2, 3	ax-mp 5	. 2 |- ( proj h ` H ) = ( x e. ~H |-> ( iota_ y e. H E. z e. ( _|_ ` H ) x = ( y +h z ) ) )
5	1, 4	fnmpti 5700	1 |- ( proj h ` H ) Fn ~H

Syntax hints:   = wceq 1374   e. wcel 1762   E.wrex 2808   |-> cmpt 4498   Fn wfn 5574   ` cfv 5579   iota_crio 6235  (class class class)co 6275   ~Hchil 25498   +h cva 25499   CHcch 25508   _|_cort 25509   proj hcpjh 25516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-hilex 25578

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-pjh 25975

This theorem is referenced by:  pjrni  26282  pjfoi  26283  pjfi  26284  dfiop2  26334  hmopidmpji  26733  pjssdif2i  26755  pjimai  26757