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Mirrors > Home > MPE Home > Th. List > dmoprabss | Structured version Visualization version GIF version |
Description: The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.) |
Ref | Expression |
---|---|
dmoprabss | ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmoprab 7506 | . 2 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} | |
2 | 19.42v 1949 | . . . 4 ⊢ (∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑧𝜑)) | |
3 | 2 | opabbii 5208 | . . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑧𝜑)} |
4 | opabssxp 5761 | . . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑧𝜑)} ⊆ (𝐴 × 𝐵) | |
5 | 3, 4 | eqsstri 4011 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵) |
6 | 1, 5 | eqsstri 4011 | 1 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∃wex 1773 ∈ wcel 2098 ⊆ wss 3943 {copab 5203 × cxp 5667 dom cdm 5669 {coprab 7406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-dm 5679 df-oprab 7409 |
This theorem is referenced by: mpondm0 7644 elmpocl 7645 oprabexd 7961 oprabex 7962 bropopvvv 8076 bropfvvvv 8078 dmaddsr 11082 dmmulsr 11083 axaddf 11142 axmulf 11143 |
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