![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > isomgrref | Structured version Visualization version GIF version |
Description: The isomorphy relation is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
Ref | Expression |
---|---|
isomgrref | ⊢ (𝐺 ∈ UHGraph → 𝐺 IsomGr 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph) | |
2 | eqid 2778 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | eqid 2778 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
4 | 2, 3 | pm3.2i 464 | . . 3 ⊢ ((Vtx‘𝐺) = (Vtx‘𝐺) ∧ (iEdg‘𝐺) = (iEdg‘𝐺)) |
5 | 4 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = (Vtx‘𝐺) ∧ (iEdg‘𝐺) = (iEdg‘𝐺))) |
6 | isomgreqve 42748 | . 2 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐺 ∈ UHGraph) ∧ ((Vtx‘𝐺) = (Vtx‘𝐺) ∧ (iEdg‘𝐺) = (iEdg‘𝐺))) → 𝐺 IsomGr 𝐺) | |
7 | 1, 1, 5, 6 | syl21anc 828 | 1 ⊢ (𝐺 ∈ UHGraph → 𝐺 IsomGr 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 Vtxcvtx 26348 iEdgciedg 26349 UHGraphcuhgr 26408 IsomGr cisomgr 42742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-uhgr 26410 df-isomgr 42744 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |