Theorem pjfni 31776

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 7319	. 2 ⊢ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)) ∈ V
2		pjfn.1	. . 3 ⊢ 𝐻 ∈ Cℋ
3		pjhfval 31471	. . 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 6635	1 ⊢ (projℎ‘𝐻) Fn ℋ

Syntax hints:   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ↦ cmpt 5179   Fn wfn 6487  ‘cfv 6492  ℩crio 7314  (class class class)co 7358   ℋchba 30994   +_ℎ cva 30995   C_ℋ cch 31004  ⊥cort 31005  proj_ℎcpjh 31012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-hilex 31074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-pjh 31470

This theorem is referenced by:  pjrni  31777  pjfoi  31778  pjfi  31779  dfiop2  31828  hmopidmpji  32227  pjssdif2i  32249  pjimai  32251