Theorem pjfni 27033

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 6244	. 2 |- ( iota_ y e. H E. z e. ( _|_ ` H ) x = ( y +h z ) ) e. _V
2		pjfn.1	. . 3 |- H e. CH
3		pjhfval 26728	. . 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 5692	1 |- ( proj h ` H ) Fn ~H

Syntax hints:   = wceq 1405   e. wcel 1842   E.wrex 2755   |-> cmpt 4453   Fn wfn 5564   ` cfv 5569   iota_crio 6239  (class class class)co 6278   ~Hchil 26250   +h cva 26251   CHcch 26260   _|_cort 26261   proj hcpjh 26268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-hilex 26330

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-pjh 26727

This theorem is referenced by:  pjrni  27034  pjfoi  27035  pjfi  27036  dfiop2  27085  hmopidmpji  27484  pjssdif2i  27506  pjimai  27508