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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 6663 | . . 3 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇)) | |
3 | 1, 2 | oveq12d 7163 | . 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 20781 | . 2 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥))) |
8 | ovex 7178 | . 2 ⊢ (𝑇𝑃( ⊥ ‘𝑇)) ∈ V | |
9 | 3, 7, 8 | fvmpt 6761 | 1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇𝑃( ⊥ ‘𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 proj1cpj1 18689 ocvcocv 20732 projcpj 20772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-pj 20775 |
This theorem is referenced by: pjf 20785 pjf2 20786 pjfo 20787 |
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