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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 5354 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | df-dm 5353 | . 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} | |
3 | vex 3418 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3418 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | brcnv 5538 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
6 | 5 | exbii 1949 | . . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦) |
7 | 6 | abbii 2945 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
8 | 1, 2, 7 | 3eqtri 2854 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∃wex 1880 {cab 2812 class class class wbr 4874 ◡ccnv 5342 dom cdm 5343 ran crn 5344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-cnv 5351 df-dm 5353 df-rn 5354 |
This theorem is referenced by: dfrn3 5545 dfdm4 5549 dm0rn0 5575 dfrnf 5598 dfima2 5710 funcnv3 6193 opabrn 29972 rncossdmcoss 34754 |
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