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Theorem pjval 20556
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 6499 . . 3 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 6994 . 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 20555 . 2 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
8 ovex 7008 . 2 (𝑇𝑃( 𝑇)) ∈ V
93, 7, 8fvmpt 6595 1 (𝑇 ∈ dom 𝐾 → (𝐾𝑇) = (𝑇𝑃( 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  dom cdm 5407  cfv 6188  (class class class)co 6976  proj1cpj1 18521  ocvcocv 20506  projcpj 20546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-map 8208  df-pj 20549
This theorem is referenced by:  pjf  20559  pjf2  20560  pjfo  20561
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