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Mirrors > Home > ILE Home > Th. List > dfdm3 | 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 4462 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
2 | df-br 3852 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 2 | exbii 1542 | . . 3 ⊢ (∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
4 | 3 | abbii 2204 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} |
5 | 1, 4 | eqtri 2109 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 ∃wex 1427 ∈ wcel 1439 {cab 2075 〈cop 3453 class class class wbr 3851 dom cdm 4452 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-11 1443 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-br 3852 df-dm 4462 |
This theorem is referenced by: csbdmg 4643 |
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