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Mirrors > Home > MPE Home > Th. List > pjval | Structured version Visualization version GIF version |
Description: Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjfval2.o | ⊢ ⊥ = (ocv‘𝑊) |
pjfval2.p | ⊢ 𝑃 = (proj1‘𝑊) |
pjfval2.k | ⊢ 𝐾 = (proj‘𝑊) |
Ref | Expression |
---|---|
pjval | ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇𝑃( ⊥ ‘𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑥 = 𝑇 → 𝑥 = 𝑇) | |
2 | fveq2 6103 | . . 3 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇)) | |
3 | 1, 2 | oveq12d 6567 | . 2 ⊢ (𝑥 = 𝑇 → (𝑥𝑃( ⊥ ‘𝑥)) = (𝑇𝑃( ⊥ ‘𝑇))) |
4 | pjfval2.o | . . 3 ⊢ ⊥ = (ocv‘𝑊) | |
5 | pjfval2.p | . . 3 ⊢ 𝑃 = (proj1‘𝑊) | |
6 | pjfval2.k | . . 3 ⊢ 𝐾 = (proj‘𝑊) | |
7 | 4, 5, 6 | pjfval2 19872 | . 2 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥))) |
8 | ovex 6577 | . 2 ⊢ (𝑇𝑃( ⊥ ‘𝑇)) ∈ V | |
9 | 3, 7, 8 | fvmpt 6191 | 1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (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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