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Mirrors > Home > MPE Home > Th. List > dfrn3 | Structured version Visualization version GIF version |
Description: Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.) |
Ref | Expression |
---|---|
dfrn3 | ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 5757 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | |
2 | df-br 5054 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 2 | exbii 1855 | . . 3 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
4 | 3 | abbii 2808 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
5 | 1, 4 | eqtri 2765 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∃wex 1787 ∈ wcel 2110 {cab 2714 〈cop 4547 class class class wbr 5053 ran crn 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-cnv 5559 df-dm 5561 df-rn 5562 |
This theorem is referenced by: elrn2g 5759 imadmrn 5939 imassrn 5940 csbrngVD 42189 |
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