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Mirrors > Home > MPE Home > Th. List > Mathboxes > grlicrel | Structured version Visualization version GIF version |
Description: The "is locally isomorphic to" relation for graphs is a relation. (Contributed by AV, 9-Jun-2025.) |
Ref | Expression |
---|---|
grlicrel | ⊢ Rel ≃𝑙𝑔𝑟 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-grlic 47884 | . . 3 ⊢ ≃𝑙𝑔𝑟 = (◡ GraphLocIso “ (V ∖ 1o)) | |
2 | cnvimass 6102 | . . . 4 ⊢ (◡ GraphLocIso “ (V ∖ 1o)) ⊆ dom GraphLocIso | |
3 | grlimfn 47882 | . . . . 5 ⊢ GraphLocIso Fn (V × V) | |
4 | 3 | fndmi 6673 | . . . 4 ⊢ dom GraphLocIso = (V × V) |
5 | 2, 4 | sseqtri 4032 | . . 3 ⊢ (◡ GraphLocIso “ (V ∖ 1o)) ⊆ (V × V) |
6 | 1, 5 | eqsstri 4030 | . 2 ⊢ ≃𝑙𝑔𝑟 ⊆ (V × V) |
7 | relxp 5707 | . 2 ⊢ Rel (V × V) | |
8 | relss 5794 | . 2 ⊢ ( ≃𝑙𝑔𝑟 ⊆ (V × V) → (Rel (V × V) → Rel ≃𝑙𝑔𝑟 )) | |
9 | 6, 7, 8 | mp2 9 | 1 ⊢ Rel ≃𝑙𝑔𝑟 |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 × cxp 5687 ◡ccnv 5688 dom cdm 5689 “ cima 5692 Rel wrel 5694 1oc1o 8498 GraphLocIso cgrlim 47879 ≃𝑙𝑔𝑟 cgrlic 47880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-f1o 6570 df-fv 6571 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-grlim 47881 df-grlic 47884 |
This theorem is referenced by: (None) |
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