Theorem pjfni 31772

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 7328	. 2 ⊢ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)) ∈ V
2		pjfn.1	. . 3 ⊢ 𝐻 ∈ Cℋ
3		pjhfval 31467	. . 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 6641	1 ⊢ (projℎ‘𝐻) Fn ℋ

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ↦ cmpt 5166   Fn wfn 6493  ‘cfv 6498  ℩crio 7323  (class class class)co 7367   ℋchba 30990   +_ℎ cva 30991   C_ℋ cch 31000  ⊥cort 31001  proj_ℎcpjh 31008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-hilex 31070

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-pjh 31466

This theorem is referenced by:  pjrni  31773  pjfoi  31774  pjfi  31775  dfiop2  31824  hmopidmpji  32223  pjssdif2i  32245  pjimai  32247