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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 6670 | . . 3 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇)) | |
3 | 1, 2 | oveq12d 7174 | . 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 20853 | . 2 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥))) |
8 | ovex 7189 | . 2 ⊢ (𝑇𝑃( ⊥ ‘𝑇)) ∈ V | |
9 | 3, 7, 8 | fvmpt 6768 | 1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇𝑃( ⊥ ‘𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 proj1cpj1 18760 ocvcocv 20804 projcpj 20844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-pj 20847 |
This theorem is referenced by: pjf 20857 pjf2 20858 pjfo 20859 |
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