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Mirrors > Home > MPE Home > Th. List > dm0rn0 | Structured version Visualization version GIF version |
Description: An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.) |
Ref | Expression |
---|---|
dm0rn0 | ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1782 | . . . . . 6 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑥∃𝑦 𝑥𝐴𝑦) | |
2 | excom 2169 | . . . . . 6 ⊢ (∃𝑥∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦∃𝑥 𝑥𝐴𝑦) | |
3 | 1, 2 | xchbinx 336 | . . . . 5 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦) |
4 | alnex 1782 | . . . . 5 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦) | |
5 | 3, 4 | bitr4i 280 | . . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦) |
6 | noel 4298 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
7 | 6 | nbn 375 | . . . . 5 ⊢ (¬ ∃𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅)) |
8 | 7 | albii 1820 | . . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅)) |
9 | noel 4298 | . . . . . 6 ⊢ ¬ 𝑦 ∈ ∅ | |
10 | 9 | nbn 375 | . . . . 5 ⊢ (¬ ∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) |
11 | 10 | albii 1820 | . . . 4 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) |
12 | 5, 8, 11 | 3bitr3i 303 | . . 3 ⊢ (∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅) ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) |
13 | abeq1 2948 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅)) | |
14 | abeq1 2948 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅ ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) | |
15 | 12, 13, 14 | 3bitr4i 305 | . 2 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅) |
16 | df-dm 5567 | . . 3 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
17 | 16 | eqeq1i 2828 | . 2 ⊢ (dom 𝐴 = ∅ ↔ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅) |
18 | dfrn2 5761 | . . 3 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | |
19 | 18 | eqeq1i 2828 | . 2 ⊢ (ran 𝐴 = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅) |
20 | 15, 17, 19 | 3bitr4i 305 | 1 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ∅c0 4293 class class class wbr 5068 dom cdm 5557 ran crn 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-cnv 5565 df-dm 5567 df-rn 5568 |
This theorem is referenced by: rn0 5798 relrn0 5842 imadisj 5950 rnsnn0 6067 f00 6563 f0rn0 6566 2nd0 7698 iinon 7979 onoviun 7982 onnseq 7983 map0b 8449 fodomfib 8800 intrnfi 8882 wdomtr 9041 noinfep 9125 wemapwe 9162 fin23lem31 9767 fin23lem40 9775 isf34lem7 9803 isf34lem6 9804 ttukeylem6 9938 fodomb 9950 rpnnen1lem4 12382 rpnnen1lem5 12383 fseqsupcl 13348 fseqsupubi 13349 dmtrclfv 14380 ruclem11 15595 prmreclem6 16259 0ram 16358 0ram2 16359 0ramcl 16361 gsumval2 17898 ghmrn 18373 gexex 18975 gsumval3 19029 subdrgint 19584 iinopn 21512 hauscmplem 22016 fbasrn 22494 alexsublem 22654 evth 23565 minveclem1 24029 minveclem3b 24033 ovollb2 24092 ovolunlem1a 24099 ovolunlem1 24100 ovoliunlem1 24105 ovoliun2 24109 ioombl1lem4 24164 uniioombllem1 24184 uniioombllem2 24186 uniioombllem6 24191 mbfsup 24267 mbfinf 24268 mbflimsup 24269 itg1climres 24317 itg2monolem1 24353 itg2mono 24356 itg2i1fseq2 24359 itg2cnlem1 24364 minvecolem1 28653 rge0scvg 31194 esumpcvgval 31339 cvmsss2 32523 fin2so 34881 ptrecube 34894 heicant 34929 isbnd3 35064 totbndbnd 35069 rnnonrel 39958 rnmpt0 41490 stoweidlem35 42327 hoicvr 42837 |
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