Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > uhgrun | Structured version Visualization version Unicode version |
Description: The union of two (undirected) hypergraphs and with the same vertex set is a hypergraph with the vertex and the union of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
Ref | Expression |
---|---|
uhgrun.g | UHGraph |
uhgrun.h | UHGraph |
uhgrun.e | iEdg |
uhgrun.f | iEdg |
uhgrun.vg | Vtx |
uhgrun.vh | Vtx |
uhgrun.i | |
uhgrun.u | |
uhgrun.v | Vtx |
uhgrun.un | iEdg |
Ref | Expression |
---|---|
uhgrun | UHGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrun.g | . . . . 5 UHGraph | |
2 | uhgrun.vg | . . . . . 6 Vtx | |
3 | uhgrun.e | . . . . . 6 iEdg | |
4 | 2, 3 | uhgrf 25957 | . . . . 5 UHGraph |
5 | 1, 4 | syl 17 | . . . 4 |
6 | uhgrun.h | . . . . . 6 UHGraph | |
7 | eqid 2622 | . . . . . . 7 Vtx Vtx | |
8 | uhgrun.f | . . . . . . 7 iEdg | |
9 | 7, 8 | uhgrf 25957 | . . . . . 6 UHGraph Vtx |
10 | 6, 9 | syl 17 | . . . . 5 Vtx |
11 | uhgrun.vh | . . . . . . . . 9 Vtx | |
12 | 11 | eqcomd 2628 | . . . . . . . 8 Vtx |
13 | 12 | pweqd 4163 | . . . . . . 7 Vtx |
14 | 13 | difeq1d 3727 | . . . . . 6 Vtx |
15 | 14 | feq3d 6032 | . . . . 5 Vtx |
16 | 10, 15 | mpbird 247 | . . . 4 |
17 | uhgrun.i | . . . 4 | |
18 | 5, 16, 17 | fun2d 6068 | . . 3 |
19 | uhgrun.un | . . . 4 iEdg | |
20 | 19 | dmeqd 5326 | . . . . 5 iEdg |
21 | dmun 5331 | . . . . 5 | |
22 | 20, 21 | syl6eq 2672 | . . . 4 iEdg |
23 | uhgrun.v | . . . . . 6 Vtx | |
24 | 23 | pweqd 4163 | . . . . 5 Vtx |
25 | 24 | difeq1d 3727 | . . . 4 Vtx |
26 | 19, 22, 25 | feq123d 6034 | . . 3 iEdg iEdgVtx |
27 | 18, 26 | mpbird 247 | . 2 iEdg iEdgVtx |
28 | uhgrun.u | . . 3 | |
29 | eqid 2622 | . . . 4 Vtx Vtx | |
30 | eqid 2622 | . . . 4 iEdg iEdg | |
31 | 29, 30 | isuhgr 25955 | . . 3 UHGraph iEdg iEdgVtx |
32 | 28, 31 | syl 17 | . 2 UHGraph iEdg iEdgVtx |
33 | 27, 32 | mpbird 247 | 1 UHGraph |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wceq 1483 wcel 1990 cdif 3571 cun 3572 cin 3573 c0 3915 cpw 4158 csn 4177 cdm 5114 wf 5884 cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 UHGraph cuhgr 25951 |
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-fv 5896 df-uhgr 25953 |
This theorem is referenced by: uhgrunop 25970 ushgrun 25971 |
Copyright terms: Public domain | W3C validator |