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Mirrors > Home > HSE Home > Th. List > pjmfn | Structured version Visualization version GIF version |
Description: Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjmfn | ⊢ projℎ Fn Cℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hilex 31022 | . . 3 ⊢ ℋ ∈ V | |
2 | 1 | mptex 7258 | . 2 ⊢ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))) ∈ V |
3 | df-pjh 31418 | . 2 ⊢ projℎ = (ℎ ∈ Cℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦)))) | |
4 | 2, 3 | fnmpti 6722 | 1 ⊢ projℎ Fn Cℋ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∃wrex 3072 ↦ cmpt 5252 Fn wfn 6567 ‘cfv 6572 ℩crio 7400 (class class class)co 7445 ℋchba 30942 +ℎ cva 30943 Cℋ cch 30952 ⊥cort 30953 projℎcpjh 30960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-hilex 31022 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-pjh 31418 |
This theorem is referenced by: pjmf1 31739 pjssdif1i 32198 dfpjop 32205 pjadj3 32211 pjcmul1i 32224 pjcmul2i 32225 |
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