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Mirrors > Home > ILE Home > Th. List > dfdm4 | GIF version |
Description: Alternate definition of domain. (Contributed by NM, 28-Dec-1996.) |
Ref | Expression |
---|---|
dfdm4 | ⊢ dom 𝐴 = ran ◡𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | vex 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 4722 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
4 | 3 | exbii 1584 | . . 3 ⊢ (∃𝑦 𝑦◡𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦) |
5 | 4 | abbii 2255 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
6 | dfrn2 4727 | . 2 ⊢ ran ◡𝐴 = {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} | |
7 | df-dm 4549 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
8 | 5, 6, 7 | 3eqtr4ri 2171 | 1 ⊢ dom 𝐴 = ran ◡𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∃wex 1468 {cab 2125 class class class wbr 3929 ◡ccnv 4538 dom cdm 4539 ran crn 4540 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: dmcnvcnv 4763 rncnvcnv 4764 rncoeq 4812 cnvimass 4902 cnvimarndm 4903 dminxp 4983 cnvsn0 5007 rnsnopg 5017 dmmpt 5034 dmco 5047 cores2 5051 cnvssrndm 5060 cocnvres 5063 unidmrn 5071 dfdm2 5073 cnvexg 5076 funimacnv 5199 foimacnv 5385 funcocnv2 5392 fimacnv 5549 f1opw2 5976 fopwdom 6730 sbthlemi4 6848 exmidfodomrlemim 7057 hmeores 12484 |
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