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Theorem pjmfn 31602
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 30886 . . 3 ℋ ∈ V
21mptex 7235 . 2 (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))) ∈ V
3 df-pjh 31282 . 2 proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))
42, 3fnmpti 6699 1 proj Fn C
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wrex 3059  cmpt 5232   Fn wfn 6544  cfv 6549  crio 7374  (class class class)co 7419  chba 30806   + cva 30807   C cch 30816  cort 30817  projcpjh 30824
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-hilex 30886
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-pjh 31282
This theorem is referenced by:  pjmf1  31603  pjssdif1i  32062  dfpjop  32069  pjadj3  32075  pjcmul1i  32088  pjcmul2i  32089
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