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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 5125 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | df-dm 5124 | . 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} | |
3 | vex 3203 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | brcnv 5305 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
6 | 5 | exbii 1774 | . . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦) |
7 | 6 | abbii 2739 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
8 | 1, 2, 7 | 3eqtri 2648 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∃wex 1704 {cab 2608 class class class wbr 4653 ◡ccnv 5113 dom cdm 5114 ran crn 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: dfrn3 5312 dfdm4 5316 dm0rn0 5342 dfrnf 5364 dfima2 5468 funcnv3 5959 opabrn 29424 |
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