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Theorem pjhfval 28585
Description: The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pjhfval (𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
Distinct variable group:   𝑥,𝑦,𝑧,𝐻

Proof of Theorem pjhfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 ( = 𝐻 = 𝐻)
2 fveq2 6353 . . . . 5 ( = 𝐻 → (⊥‘) = (⊥‘𝐻))
32rexeqdv 3284 . . . 4 ( = 𝐻 → (∃𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦)))
41, 3riotaeqbidv 6778 . . 3 ( = 𝐻 → (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦)) = (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦)))
54mpteq2dv 4897 . 2 ( = 𝐻 → (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
6 df-pjh 28584 . 2 proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))
7 ax-hilex 28186 . . 3 ℋ ∈ V
87mptex 6651 . 2 (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))) ∈ V
95, 6, 8fvmpt 6445 1 (𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wrex 3051  cmpt 4881  cfv 6049  crio 6774  (class class class)co 6814  chil 28106   + cva 28107   C cch 28116  cort 28117  projcpjh 28124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-hilex 28186
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-pjh 28584
This theorem is referenced by:  pjhval  28586  pjfni  28890
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