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Mirrors > Home > ILE Home > Th. List > dfrn2 | GIF version |
Description: Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.) |
Ref | Expression |
---|---|
dfrn2 | ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 4382 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | df-dm 4381 | . 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} | |
3 | vex 2605 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 2605 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | brcnv 4546 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
6 | 5 | exbii 1537 | . . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦) |
7 | 6 | abbii 2195 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
8 | 1, 2, 7 | 3eqtri 2106 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 ∃wex 1422 {cab 2068 class class class wbr 3793 ◡ccnv 4370 dom cdm 4371 ran crn 4372 |
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 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-v 2604 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-br 3794 df-opab 3848 df-cnv 4379 df-dm 4381 df-rn 4382 |
This theorem is referenced by: dfrn3 4552 dfdm4 4555 dm0rn0 4580 dmmrnm 4582 dfrnf 4603 dfima2 4700 funcnv3 4992 |
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