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| 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 5695 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
| 2 | df-br 5144 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
| 3 | 2 | exbii 1848 | . . 3 ⊢ (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | 
| 4 | 3 | abbii 2809 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} | 
| 5 | 1, 4 | eqtri 2765 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 〈cop 4632 class class class wbr 5143 dom cdm 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-br 5144 df-dm 5695 | 
| This theorem is referenced by: csbdm 5908 cnextf 24074 dmrab 32516 | 
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