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Theorem pjmfn 31462
Description: Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
pjmfn proj Fn C

Proof of Theorem pjmfn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 30746 . . 3 ℋ ∈ V
21mptex 7217 . 2 (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))) ∈ V
3 df-pjh 31142 . 2 proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))
42, 3fnmpti 6684 1 proj Fn C
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wrex 3062  cmpt 5222   Fn wfn 6529  cfv 6534  crio 7357  (class class class)co 7402  chba 30666   + cva 30667   C cch 30676  cort 30677  projcpjh 30684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-hilex 30746
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-pjh 31142
This theorem is referenced by:  pjmf1  31463  pjssdif1i  31922  dfpjop  31929  pjadj3  31935  pjcmul1i  31948  pjcmul2i  31949
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