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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 2733 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | vex 2733 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 4792 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
4 | 3 | exbii 1598 | . . 3 ⊢ (∃𝑦 𝑦◡𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦) |
5 | 4 | abbii 2286 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
6 | dfrn2 4797 | . 2 ⊢ ran ◡𝐴 = {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} | |
7 | df-dm 4619 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
8 | 5, 6, 7 | 3eqtr4ri 2202 | 1 ⊢ dom 𝐴 = ran ◡𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∃wex 1485 {cab 2156 class class class wbr 3987 ◡ccnv 4608 dom cdm 4609 ran crn 4610 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-cnv 4617 df-dm 4619 df-rn 4620 |
This theorem is referenced by: dmcnvcnv 4833 rncnvcnv 4834 rncoeq 4882 cnvimass 4972 cnvimarndm 4973 dminxp 5053 cnvsn0 5077 rnsnopg 5087 dmmpt 5104 dmco 5117 cores2 5121 cnvssrndm 5130 cocnvres 5133 unidmrn 5141 dfdm2 5143 cnvexg 5146 funimacnv 5272 foimacnv 5458 funcocnv2 5465 fimacnv 5622 f1opw2 6052 fopwdom 6810 sbthlemi4 6933 exmidfodomrlemim 7165 hmeores 13068 |
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