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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 4298 | . 2 ⊢ dom A = {x ∣ ∃y xAy} | |
2 | df-br 3756 | . . . 4 ⊢ (xAy ↔ 〈x, y〉 ∈ A) | |
3 | 2 | exbii 1493 | . . 3 ⊢ (∃y xAy ↔ ∃y〈x, y〉 ∈ A) |
4 | 3 | abbii 2150 | . 2 ⊢ {x ∣ ∃y xAy} = {x ∣ ∃y〈x, y〉 ∈ A} |
5 | 1, 4 | eqtri 2057 | 1 ⊢ dom A = {x ∣ ∃y〈x, y〉 ∈ A} |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∃wex 1378 ∈ wcel 1390 {cab 2023 〈cop 3370 class class class wbr 3755 dom cdm 4288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-br 3756 df-dm 4298 |
This theorem is referenced by: csbdmg 4472 |
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