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Theorem pjhfval 29179
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 6645 . . . . 5 ( = 𝐻 → (⊥‘) = (⊥‘𝐻))
32rexeqdv 3365 . . . 4 ( = 𝐻 → (∃𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦)))
41, 3riotaeqbidv 7096 . . 3 ( = 𝐻 → (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦)) = (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦)))
54mpteq2dv 5126 . 2 ( = 𝐻 → (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
6 df-pjh 29178 . 2 proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))
7 ax-hilex 28782 . . 3 ℋ ∈ V
87mptex 6963 . 2 (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))) ∈ V
95, 6, 8fvmpt 6745 1 (𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  wrex 3107  cmpt 5110  cfv 6324  crio 7092  (class class class)co 7135  chba 28702   + cva 28703   C cch 28712  cort 28713  projcpjh 28720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-hilex 28782
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-pjh 29178
This theorem is referenced by:  pjhval  29180  pjfni  29484
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