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Theorem pjval 21132
Description: Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval2.o βŠ₯ = (ocvβ€˜π‘Š)
pjfval2.p 𝑃 = (proj1β€˜π‘Š)
pjfval2.k 𝐾 = (projβ€˜π‘Š)
Assertion
Ref Expression
pjval (𝑇 ∈ dom 𝐾 β†’ (πΎβ€˜π‘‡) = (𝑇𝑃( βŠ₯ β€˜π‘‡)))

Proof of Theorem pjval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (π‘₯ = 𝑇 β†’ π‘₯ = 𝑇)
2 fveq2 6843 . . 3 (π‘₯ = 𝑇 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘‡))
31, 2oveq12d 7376 . 2 (π‘₯ = 𝑇 β†’ (π‘₯𝑃( βŠ₯ β€˜π‘₯)) = (𝑇𝑃( βŠ₯ β€˜π‘‡)))
4 pjfval2.o . . 3 βŠ₯ = (ocvβ€˜π‘Š)
5 pjfval2.p . . 3 𝑃 = (proj1β€˜π‘Š)
6 pjfval2.k . . 3 𝐾 = (projβ€˜π‘Š)
74, 5, 6pjfval2 21131 . 2 𝐾 = (π‘₯ ∈ dom 𝐾 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯)))
8 ovex 7391 . 2 (𝑇𝑃( βŠ₯ β€˜π‘‡)) ∈ V
93, 7, 8fvmpt 6949 1 (𝑇 ∈ dom 𝐾 β†’ (πΎβ€˜π‘‡) = (𝑇𝑃( βŠ₯ β€˜π‘‡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  dom cdm 5634  β€˜cfv 6497  (class class class)co 7358  proj1cpj1 19422  ocvcocv 21080  projcpj 21122
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-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-pj 21125
This theorem is referenced by:  pjf  21135  pjf2  21136  pjfo  21137
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