Theorem pjfni 25257

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 6166	. 2 |- ( iota_ y e. H E. z e. ( _|_ ` H ) x = ( y +h z ) ) e. _V
2		pjfn.1	. . 3 |- H e. CH
3		pjhfval 24952	. . 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 5648	1 |- ( proj h ` H ) Fn ~H

Syntax hints:   = wceq 1370   e. wcel 1758   E.wrex 2800   |-> cmpt 4459   Fn wfn 5522   ` cfv 5527   iota_crio 6161  (class class class)co 6201   ~Hchil 24474   +h cva 24475   CHcch 24484   _|_cort 24485   proj hcpjh 24492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-hilex 24554

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-pjh 24951

This theorem is referenced by:  pjrni  25258  pjfoi  25259  pjfi  25260  dfiop2  25310  hmopidmpji  25709  pjssdif2i  25731  pjimai  25733