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Mirrors > Home > HSE Home > Th. List > pjfni | Structured version Visualization version GIF version |
Description: Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjfn.1 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjfni | ⊢ (projℎ‘𝐻) Fn ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaex 7104 | . 2 ⊢ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)) ∈ V | |
2 | pjfn.1 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
3 | pjhfval 29157 | . . 3 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)))) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧))) |
5 | 1, 4 | fnmpti 6477 | 1 ⊢ (projℎ‘𝐻) Fn ℋ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ↦ cmpt 5132 Fn wfn 6336 ‘cfv 6341 ℩crio 7099 (class class class)co 7142 ℋchba 28680 +ℎ cva 28681 Cℋ cch 28690 ⊥cort 28691 projℎcpjh 28698 |
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-rep 5176 ax-sep 5189 ax-nul 5196 ax-pr 5316 ax-hilex 28760 |
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-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-pjh 29156 |
This theorem is referenced by: pjrni 29463 pjfoi 29464 pjfi 29465 dfiop2 29514 hmopidmpji 29913 pjssdif2i 29935 pjimai 29937 |
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