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Mirrors > Home > MPE Home > Th. List > dfdm4 | Structured version Visualization version 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 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 5227 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
4 | 3 | exbii 1764 | . . 3 ⊢ (∃𝑦 𝑦◡𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦) |
5 | 4 | abbii 2726 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
6 | dfrn2 5233 | . 2 ⊢ ran ◡𝐴 = {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} | |
7 | df-dm 5048 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
8 | 5, 6, 7 | 3eqtr4ri 2643 | 1 ⊢ dom 𝐴 = ran ◡𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∃wex 1695 {cab 2596 class class class wbr 4583 ◡ccnv 5037 dom cdm 5038 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: dmcnvcnv 5269 rncnvcnv 5270 rncoeq 5310 cnvimass 5404 cnvimarndm 5405 dminxp 5493 cnvsn0 5521 rnsnopg 5532 dmmpt 5547 dmco 5560 cores2 5565 cnvssrndm 5574 unidmrn 5582 dfdm2 5584 funimacnv 5884 foimacnv 6067 funcocnv2 6074 fimacnv 6255 f1opw2 6786 cnvexg 7005 tz7.48-3 7426 fopwdom 7953 sbthlem4 7958 fodomr 7996 f1opwfi 8153 zorn2lem4 9204 trclublem 13582 relexpcnv 13623 unbenlem 15450 gsumpropd2lem 17096 pjdm 19870 paste 20908 hmeores 21384 icchmeo 22548 fcnvgreu 28855 ffsrn 28892 gsummpt2co 29111 coinfliprv 29871 itg2addnclem2 32632 lnmlmic 36676 dmnonrel 36915 cnvrcl0 36951 conrel1d 36974 |
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