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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 2554 | . . . . 5 ⊢ y ∈ V | |
2 | vex 2554 | . . . . 5 ⊢ x ∈ V | |
3 | 1, 2 | brcnv 4461 | . . . 4 ⊢ (y◡Ax ↔ xAy) |
4 | 3 | exbii 1493 | . . 3 ⊢ (∃y y◡Ax ↔ ∃y xAy) |
5 | 4 | abbii 2150 | . 2 ⊢ {x ∣ ∃y y◡Ax} = {x ∣ ∃y xAy} |
6 | dfrn2 4466 | . 2 ⊢ ran ◡A = {x ∣ ∃y y◡Ax} | |
7 | df-dm 4298 | . 2 ⊢ dom A = {x ∣ ∃y xAy} | |
8 | 5, 6, 7 | 3eqtr4ri 2068 | 1 ⊢ dom A = ran ◡A |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∃wex 1378 {cab 2023 class class class wbr 3755 ◡ccnv 4287 dom cdm 4288 ran crn 4289 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-cnv 4296 df-dm 4298 df-rn 4299 |
This theorem is referenced by: dmcnvcnv 4501 rncnvcnv 4502 rncoeq 4548 cnvimass 4631 cnvimarndm 4632 dminxp 4708 cnvsn0 4732 rnsnopg 4742 dmmpt 4759 dmco 4772 cores2 4776 cnvssrndm 4785 unidmrn 4793 dfdm2 4795 cnvexg 4798 funimacnv 4918 foimacnv 5087 funcocnv2 5094 fimacnv 5239 f1opw2 5648 fopwdom 6246 |
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