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Theorem pjval 19873
 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 6103 . . 3 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 6567 . 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 19872 . 2 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
8 ovex 6577 . 2 (𝑇𝑃( 𝑇)) ∈ V
93, 7, 8fvmpt 6191 1 (𝑇 ∈ dom 𝐾 → (𝐾𝑇) = (𝑇𝑃( 𝑇)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  proj1cpj1 17873  ocvcocv 19823  projcpj 19863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-pj 19866 This theorem is referenced by:  pjf  19876  pjf2  19877  pjfo  19878
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