Theorem pjfni 32360

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 7412	. 2 ⊢ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)) ∈ V
2		pjfn.1	. . 3 ⊢ 𝐻 ∈ Cℋ
3		pjhfval 32055	. . 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 6712	1 ⊢ (projℎ‘𝐻) Fn ℋ

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ↦ cmpt 5192   Fn wfn 6561  ‘cfv 6566  ℩crio 7407  (class class class)co 7451   ℋchba 31578   +_ℎ cva 31579   C_ℋ cch 31588  ⊥cort 31589  proj_ℎcpjh 31596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-hilex 31658

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-pjh 32054

This theorem is used by:  pjrni  32361  pjfoi  32362  pjfi  32363  dfiop2  32412  hmopidmpji  32811  pjssdif2i  32833  pjimai  32835