Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4545 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | df-dm 4544 | . 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} | |
3 | vex 2684 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 2684 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | brcnv 4717 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
6 | 5 | exbii 1584 | . . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦) |
7 | 6 | abbii 2253 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
8 | 1, 2, 7 | 3eqtri 2162 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∃wex 1468 {cab 2123 class class class wbr 3924 ◡ccnv 4533 dom cdm 4534 ran crn 4535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-cnv 4542 df-dm 4544 df-rn 4545 |
This theorem is referenced by: dfrn3 4723 dfdm4 4726 dm0rn0 4751 dmmrnm 4753 dfrnf 4775 dfima2 4878 funcnv3 5180 |
Copyright terms: Public domain | W3C validator |