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Mirrors > Home > HSE Home > Th. List > pjhfval | Structured version Visualization version GIF version |
Description: The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjhfval | ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (ℎ = 𝐻 → ℎ = 𝐻) | |
2 | fveq2 6695 | . . . . 5 ⊢ (ℎ = 𝐻 → (⊥‘ℎ) = (⊥‘𝐻)) | |
3 | 2 | rexeqdv 3316 | . . . 4 ⊢ (ℎ = 𝐻 → (∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))) |
4 | 1, 3 | riotaeqbidv 7151 | . . 3 ⊢ (ℎ = 𝐻 → (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦)) = (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))) |
5 | 4 | mpteq2dv 5136 | . 2 ⊢ (ℎ = 𝐻 → (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))) = (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦)))) |
6 | df-pjh 29430 | . 2 ⊢ projℎ = (ℎ ∈ Cℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦)))) | |
7 | ax-hilex 29034 | . . 3 ⊢ ℋ ∈ V | |
8 | 7 | mptex 7017 | . 2 ⊢ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))) ∈ V |
9 | 5, 6, 8 | fvmpt 6796 | 1 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 ↦ cmpt 5120 ‘cfv 6358 ℩crio 7147 (class class class)co 7191 ℋchba 28954 +ℎ cva 28955 Cℋ cch 28964 ⊥cort 28965 projℎcpjh 28972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-hilex 29034 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-pjh 29430 |
This theorem is referenced by: pjhval 29432 pjfni 29736 |
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