Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mapdm0 | Structured version Visualization version Unicode version |
Description: The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.) |
Ref | Expression |
---|---|
mapdm0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4790 | . . . . 5 | |
2 | elmapg 7870 | . . . . 5 | |
3 | 1, 2 | mpan2 707 | . . . 4 |
4 | f0bi 6088 | . . . 4 | |
5 | 3, 4 | syl6bb 276 | . . 3 |
6 | vex 3203 | . . . 4 | |
7 | 6 | elsn 4192 | . . 3 |
8 | 5, 7 | syl6bbr 278 | . 2 |
9 | 8 | eqrdv 2620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wceq 1483 wcel 1990 cvv 3200 c0 3915 csn 4177 wf 5884 (class class class)co 6650 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 |
This theorem is referenced by: repr0 30689 mpct 39393 rrxtopn0 40513 qndenserrnbl 40515 hoicvr 40762 ovn02 40782 ovnhoi 40817 ovnlecvr2 40824 hoiqssbl 40839 hoimbl 40845 |
Copyright terms: Public domain | W3C validator |