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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 2754 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | vex 2754 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 4824 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
4 | 3 | exbii 1615 | . . 3 ⊢ (∃𝑦 𝑦◡𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦) |
5 | 4 | abbii 2304 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
6 | dfrn2 4829 | . 2 ⊢ ran ◡𝐴 = {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} | |
7 | df-dm 4650 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
8 | 5, 6, 7 | 3eqtr4ri 2220 | 1 ⊢ dom 𝐴 = ran ◡𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∃wex 1502 {cab 2174 class class class wbr 4017 ◡ccnv 4639 dom cdm 4640 ran crn 4641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-v 2753 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-br 4018 df-opab 4079 df-cnv 4648 df-dm 4650 df-rn 4651 |
This theorem is referenced by: dmcnvcnv 4865 rncnvcnv 4866 rncoeq 4914 cnvimass 5005 cnvimarndm 5006 dminxp 5087 cnvsn0 5111 rnsnopg 5121 dmmpt 5138 dmco 5151 cores2 5155 cnvssrndm 5164 cocnvres 5167 unidmrn 5175 dfdm2 5177 cnvexg 5180 funimacnv 5306 foimacnv 5493 funcocnv2 5500 fimacnv 5660 f1opw2 6094 fopwdom 6853 sbthlemi4 6976 exmidfodomrlemim 7217 hmeores 14198 |
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