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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 2726 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | eqid 2726 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
4 | 2, 3 | pm3.2i 470 | . . 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 47065 | . 2 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐺 ∈ UHGraph) ∧ ((Vtx‘𝐺) = (Vtx‘𝐺) ∧ (iEdg‘𝐺) = (iEdg‘𝐺))) → 𝐺 IsomGr 𝐺) | |
7 | 1, 1, 5, 6 | syl21anc 835 | 1 ⊢ (𝐺 ∈ UHGraph → 𝐺 IsomGr 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ‘cfv 6537 Vtxcvtx 28764 iEdgciedg 28765 UHGraphcuhgr 28824 IsomGr cisomgr 47059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-uhgr 28826 df-isomgr 47061 |
This theorem is referenced by: (None) |
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