Theorem pjfni 29964

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 ∈ Cℋ

Assertion

Ref	Expression
pjfni	⊢ (projℎ‘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 7216	. 2 ⊢ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)) ∈ V
2		pjfn.1	. . 3 ⊢ 𝐻 ∈ Cℋ
3		pjhfval 29659	. . 3 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧))))
4	2, 3	ax-mp 5	. 2 ⊢ (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)))
5	1, 4	fnmpti 6560	1 ⊢ (projℎ‘𝐻) Fn ℋ

Syntax hints:   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ↦ cmpt 5153   Fn wfn 6413  ‘cfv 6418  ℩crio 7211  (class class class)co 7255   ℋchba 29182   +_ℎ cva 29183   C_ℋ cch 29192  ⊥cort 29193  proj_ℎcpjh 29200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-pjh 29658

This theorem is referenced by:  pjrni  29965  pjfoi  29966  pjfi  29967  dfiop2  30016  hmopidmpji  30415  pjssdif2i  30437  pjimai  30439