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Mirrors > Home > MPE Home > Th. List > dfrn2 | Structured version Visualization version 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 5049 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | df-dm 5048 | . 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} | |
3 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | brcnv 5227 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
6 | 5 | exbii 1764 | . . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦) |
7 | 6 | abbii 2726 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
8 | 1, 2, 7 | 3eqtri 2636 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∃wex 1695 {cab 2596 class class class wbr 4583 ◡ccnv 5037 dom cdm 5038 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: dfrn3 5234 dfdm4 5238 dm0rn0 5263 dfrnf 5285 dfima2 5387 funcnv3 5873 opabrn 28804 |
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