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Theorem pjval 20402
 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 6649 . . 3 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 7157 . 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 20401 . 2 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
8 ovex 7172 . 2 (𝑇𝑃( 𝑇)) ∈ V
93, 7, 8fvmpt 6749 1 (𝑇 ∈ dom 𝐾 → (𝐾𝑇) = (𝑇𝑃( 𝑇)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  dom cdm 5523  ‘cfv 6328  (class class class)co 7139  proj1cpj1 18755  ocvcocv 20352  projcpj 20392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-pj 20395 This theorem is referenced by:  pjf  20405  pjf2  20406  pjfo  20407
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