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Theorem pjmfn 31738
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 31022 . . 3 ℋ ∈ V
21mptex 7258 . 2 (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))) ∈ V
3 df-pjh 31418 . 2 proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))
42, 3fnmpti 6722 1 proj Fn C
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wrex 3072  cmpt 5252   Fn wfn 6567  cfv 6572  crio 7400  (class class class)co 7445  chba 30942   + cva 30943   C cch 30952  cort 30953  projcpjh 30960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-hilex 31022
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-pjh 31418
This theorem is referenced by:  pjmf1  31739  pjssdif1i  32198  dfpjop  32205  pjadj3  32211  pjcmul1i  32224  pjcmul2i  32225
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