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Theorem pjval 20782
Description: Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval2.o = (ocv‘𝑊)
pjfval2.p 𝑃 = (proj1𝑊)
pjfval2.k 𝐾 = (proj‘𝑊)
Assertion
Ref Expression
pjval (𝑇 ∈ dom 𝐾 → (𝐾𝑇) = (𝑇𝑃( 𝑇)))

Proof of Theorem pjval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝑥 = 𝑇𝑥 = 𝑇)
2 fveq2 6663 . . 3 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 7163 . 2 (𝑥 = 𝑇 → (𝑥𝑃( 𝑥)) = (𝑇𝑃( 𝑇)))
4 pjfval2.o . . 3 = (ocv‘𝑊)
5 pjfval2.p . . 3 𝑃 = (proj1𝑊)
6 pjfval2.k . . 3 𝐾 = (proj‘𝑊)
74, 5, 6pjfval2 20781 . 2 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
8 ovex 7178 . 2 (𝑇𝑃( 𝑇)) ∈ V
93, 7, 8fvmpt 6761 1 (𝑇 ∈ dom 𝐾 → (𝐾𝑇) = (𝑇𝑃( 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  dom cdm 5548  cfv 6348  (class class class)co 7145  proj1cpj1 18689  ocvcocv 20732  projcpj 20772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-pj 20775
This theorem is referenced by:  pjf  20785  pjf2  20786  pjfo  20787
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