Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > uspgrun | Structured version Visualization version Unicode version |
Description: The union of two simple pseudographs and with the same vertex set is a pseudograph with the vertex and the union of the (indexed) edges. (Contributed by AV, 16-Oct-2020.) |
Ref | Expression |
---|---|
uspgrun.g | USPGraph |
uspgrun.h | USPGraph |
uspgrun.e | iEdg |
uspgrun.f | iEdg |
uspgrun.vg | Vtx |
uspgrun.vh | Vtx |
uspgrun.i | |
uspgrun.u | |
uspgrun.v | Vtx |
uspgrun.un | iEdg |
Ref | Expression |
---|---|
uspgrun | UPGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrun.g | . . 3 USPGraph | |
2 | uspgrupgr 26071 | . . 3 USPGraph UPGraph | |
3 | 1, 2 | syl 17 | . 2 UPGraph |
4 | uspgrun.h | . . 3 USPGraph | |
5 | uspgrupgr 26071 | . . 3 USPGraph UPGraph | |
6 | 4, 5 | syl 17 | . 2 UPGraph |
7 | uspgrun.e | . 2 iEdg | |
8 | uspgrun.f | . 2 iEdg | |
9 | uspgrun.vg | . 2 Vtx | |
10 | uspgrun.vh | . 2 Vtx | |
11 | uspgrun.i | . 2 | |
12 | uspgrun.u | . 2 | |
13 | uspgrun.v | . 2 Vtx | |
14 | uspgrun.un | . 2 iEdg | |
15 | 3, 6, 7, 8, 9, 10, 11, 12, 13, 14 | upgrun 26013 | 1 UPGraph |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cun 3572 cin 3573 c0 3915 cdm 5114 cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 UPGraph cupgr 25975 USPGraph cuspgr 26043 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-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-f1 5893 df-fv 5896 df-upgr 25977 df-uspgr 26045 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |