Theorem pjmfn 27958

Description: Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
pjmfn	⊢ projℎ Fn Cℋ

Proof of Theorem pjmfn

Dummy variables 𝑥 ℎ 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 27240	. . 3 ⊢ ℋ ∈ V
2	1	mptex 6390	. 2 ⊢ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))) ∈ V
3		df-pjh 27638	. 2 ⊢ projℎ = (ℎ ∈ Cℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))))
4	2, 3	fnmpti 5935	1 ⊢ projℎ Fn Cℋ

Syntax hints:   = wceq 1475  ∃wrex 2897   ↦ cmpt 4643   Fn wfn 5799  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   ℋchil 27160   +_ℎ cva 27161   C_ℋ cch 27170  ⊥cort 27171  proj_ℎcpjh 27178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-hilex 27240

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-pjh 27638

This theorem is referenced by:  pjmf1  27959  pjssdif1i  28418  dfpjop  28425  pjadj3  28431  pjcmul1i  28444  pjcmul2i  28445