Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5048 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
2 | df-br 4584 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 2 | exbii 1764 | . . 3 ⊢ (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
4 | 3 | abbii 2726 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} |
5 | 1, 4 | eqtri 2632 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 〈cop 4131 class class class wbr 4583 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-br 4584 df-dm 5048 |
This theorem is referenced by: csbdm 5240 cnextf 21680 |
Copyright terms: Public domain | W3C validator |