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Mirrors > Home > MPE Home > Th. List > dfdm3 | Structured version Visualization version GIF version |
Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.) |
Ref | Expression |
---|---|
dfdm3 | ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dm 5691 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
2 | df-br 5153 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 2 | exbii 1842 | . . 3 ⊢ (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
4 | 3 | abbii 2795 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} |
5 | 1, 4 | eqtri 2753 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2702 〈cop 4638 class class class wbr 5152 dom cdm 5681 |
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-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-br 5153 df-dm 5691 |
This theorem is referenced by: csbdm 5903 cnextf 24053 dmrab 32416 |
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