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Mirrors > Home > HSE Home > Th. List > pjhval | Structured version Visualization version GIF version |
Description: Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjhval | ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjhfval 31425 | . . 3 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))) | |
2 | 1 | fveq1d 6909 | . 2 ⊢ (𝐻 ∈ Cℋ → ((projℎ‘𝐻)‘𝐴) = ((𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))‘𝐴)) |
3 | eqeq1 2739 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 +ℎ 𝑦) ↔ 𝐴 = (𝑥 +ℎ 𝑦))) | |
4 | 3 | rexbidv 3177 | . . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
5 | 4 | riotabidv 7390 | . . 3 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
6 | eqid 2735 | . . 3 ⊢ (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))) | |
7 | riotaex 7392 | . . 3 ⊢ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) ∈ V | |
8 | 5, 6, 7 | fvmpt 7016 | . 2 ⊢ (𝐴 ∈ ℋ → ((𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
9 | 2, 8 | sylan9eq 2795 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 ℋchba 30948 +ℎ cva 30949 Cℋ cch 30958 ⊥cort 30959 projℎcpjh 30966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-hilex 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-pjh 31424 |
This theorem is referenced by: pjpreeq 31427 |
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