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Mirrors > Home > MPE Home > Th. List > Mathboxes > grlicer | Structured version Visualization version GIF version |
Description: Local isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 11-Jun-2025.) |
Ref | Expression |
---|---|
grlicer | ⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grlicref 47831 | . 2 ⊢ (𝑓 ∈ UHGraph → 𝑓 ≃𝑙𝑔𝑟 𝑓) | |
2 | grlicsym 47832 | . 2 ⊢ (𝑓 ∈ UHGraph → (𝑓 ≃𝑙𝑔𝑟 𝑔 → 𝑔 ≃𝑙𝑔𝑟 𝑓)) | |
3 | grlictr 47834 | . . 3 ⊢ ((𝑓 ≃𝑙𝑔𝑟 𝑔 ∧ 𝑔 ≃𝑙𝑔𝑟 ℎ) → 𝑓 ≃𝑙𝑔𝑟 ℎ) | |
4 | 3 | a1i 11 | . 2 ⊢ (𝑓 ∈ UHGraph → ((𝑓 ≃𝑙𝑔𝑟 𝑔 ∧ 𝑔 ≃𝑙𝑔𝑟 ℎ) → 𝑓 ≃𝑙𝑔𝑟 ℎ)) |
5 | 1, 2, 4 | brinxper 8794 | 1 ⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∩ cin 3975 class class class wbr 5166 × cxp 5698 Er wer 8762 UHGraphcuhgr 29093 ≃𝑙𝑔𝑟 cgrlic 47803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1st 8032 df-2nd 8033 df-1o 8524 df-er 8765 df-map 8888 df-vtx 29035 df-iedg 29036 df-uhgr 29095 df-clnbgr 47695 df-isubgr 47735 df-grim 47750 df-gric 47753 df-grlim 47804 df-grlic 47807 |
This theorem is referenced by: (None) |
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