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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 3449 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | vex 3449 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 1, 2 | brcnv 5838 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
4 | 3 | exbii 1850 | . . 3 ⊢ (∃𝑦 𝑦◡𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦) |
5 | 4 | abbii 2806 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
6 | dfrn2 5844 | . 2 ⊢ ran ◡𝐴 = {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} | |
7 | df-dm 5643 | . 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
8 | 5, 6, 7 | 3eqtr4ri 2775 | 1 ⊢ dom 𝐴 = ran ◡𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∃wex 1781 {cab 2713 class class class wbr 5105 ◡ccnv 5632 dom cdm 5633 ran crn 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-cnv 5641 df-dm 5643 df-rn 5644 |
This theorem is referenced by: dmcnvcnv 5888 rncnvcnv 5889 rncoeq 5930 cnvimass 6033 cnvimarndm 6034 dminxp 6132 cnvsn0 6162 rnsnopg 6173 dmmpt 6192 dmco 6206 cores2 6211 cnvssrndm 6223 unidmrn 6231 dfdm2 6233 funimacnv 6582 foimacnv 6801 funcocnv2 6809 fimacnvOLD 7021 f1opw2 7608 cnvexg 7861 tz7.48-3 8390 fopwdom 9024 sbthlem4 9030 fodomr 9072 cnvfi 9124 f1opwfi 9300 zorn2lem4 10435 trclublem 14880 relexpcnv 14920 unbenlem 16780 gsumpropd2lem 18534 pjdm 21113 paste 22645 hmeores 23122 icchmeo 24304 fcnvgreu 31589 ffsrn 31646 gsummpt2co 31890 tocycfvres1 31959 tocycfvres2 31960 cycpmfvlem 31961 cycpmfv3 31964 coinfliprv 33082 itg2addnclem2 36130 rncnv 36761 lnmlmic 41401 dmnonrel 41852 cnvrcl0 41887 conrel1d 41925 |
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