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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 5699 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
2 | df-br 5149 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 2 | exbii 1845 | . . 3 ⊢ (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
4 | 3 | abbii 2807 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} |
5 | 1, 4 | eqtri 2763 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 〈cop 4637 class class class wbr 5148 dom cdm 5689 |
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-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-br 5149 df-dm 5699 |
This theorem is referenced by: csbdm 5911 cnextf 24090 dmrab 32525 |
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