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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 29766 | . . 3 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))) | |
2 | 1 | fveq1d 6768 | . 2 ⊢ (𝐻 ∈ Cℋ → ((projℎ‘𝐻)‘𝐴) = ((𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))‘𝐴)) |
3 | eqeq1 2742 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 +ℎ 𝑦) ↔ 𝐴 = (𝑥 +ℎ 𝑦))) | |
4 | 3 | rexbidv 3224 | . . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
5 | 4 | riotabidv 7226 | . . 3 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
6 | eqid 2738 | . . 3 ⊢ (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))) | |
7 | riotaex 7228 | . . 3 ⊢ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6867 | . 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 2798 | 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 396 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 ↦ cmpt 5156 ‘cfv 6426 ℩crio 7223 (class class class)co 7267 ℋchba 29289 +ℎ cva 29290 Cℋ cch 29299 ⊥cort 29300 projℎcpjh 29307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pr 5350 ax-hilex 29369 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-pjh 29765 |
This theorem is referenced by: pjpreeq 29768 |
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