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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 6905 | . . 3 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇)) | |
| 3 | 1, 2 | oveq12d 7450 | . 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 21730 | . 2 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥))) | 
| 8 | ovex 7465 | . 2 ⊢ (𝑇𝑃( ⊥ ‘𝑇)) ∈ V | |
| 9 | 3, 7, 8 | fvmpt 7015 | 1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇𝑃( ⊥ ‘𝑇))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 dom cdm 5684 ‘cfv 6560 (class class class)co 7432 proj1cpj1 19654 ocvcocv 21679 projcpj 21721 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-pj 21724 | 
| This theorem is referenced by: pjf 21734 pjf2 21735 pjfo 21736 | 
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