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Mirrors > Home > ILE Home > Th. List > dfrn3 | GIF version |
Description: Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.) |
Ref | Expression |
---|---|
dfrn3 | ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 4541 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | |
2 | df-br 3786 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 2 | exbii 1536 | . . 3 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
4 | 3 | abbii 2194 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
5 | 1, 4 | eqtri 2101 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∃wex 1421 ∈ wcel 1433 {cab 2067 〈cop 3401 class class class wbr 3785 ran crn 4364 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: elrn2g 4543 elrn2 4594 imadmrn 4698 imassrn 4699 csbrng 4802 |
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