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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 2604 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | vex 2604 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 4536 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
4 | 3 | exbii 1536 | . . 3 ⊢ (∃𝑦 𝑦◡𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦) |
5 | 4 | abbii 2194 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
6 | dfrn2 4541 | . 2 ⊢ ran ◡𝐴 = {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} | |
7 | df-dm 4373 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
8 | 5, 6, 7 | 3eqtr4ri 2112 | 1 ⊢ dom 𝐴 = ran ◡𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∃wex 1421 {cab 2067 class class class wbr 3785 ◡ccnv 4362 dom cdm 4363 ran crn 4364 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: dmcnvcnv 4576 rncnvcnv 4577 rncoeq 4623 cnvimass 4708 cnvimarndm 4709 dminxp 4785 cnvsn0 4809 rnsnopg 4819 dmmpt 4836 dmco 4849 cores2 4853 cnvssrndm 4862 unidmrn 4870 dfdm2 4872 cnvexg 4875 funimacnv 4995 foimacnv 5164 funcocnv2 5171 fimacnv 5317 f1opw2 5726 fopwdom 6333 |
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