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Theorem pjhfval 30626
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 6887 . . . . 5 ( = 𝐻 → (⊥‘) = (⊥‘𝐻))
32rexeqdv 3327 . . . 4 ( = 𝐻 → (∃𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦)))
41, 3riotaeqbidv 7362 . . 3 ( = 𝐻 → (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦)) = (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦)))
54mpteq2dv 5248 . 2 ( = 𝐻 → (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
6 df-pjh 30625 . 2 proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))
7 ax-hilex 30229 . . 3 ℋ ∈ V
87mptex 7219 . 2 (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))) ∈ V
95, 6, 8fvmpt 6993 1 (𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wrex 3071  cmpt 5229  cfv 6539  crio 7358  (class class class)co 7403  chba 30149   + cva 30150   C cch 30159  cort 30160  projcpjh 30167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pr 5425  ax-hilex 30229
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-pjh 30625
This theorem is referenced by:  pjhval  30627  pjfni  30931
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