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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 47948 | . . 3 ⊢ ≃𝑙𝑔𝑟 = (◡ GraphLocIso “ (V ∖ 1o)) | |
| 2 | cnvimass 6100 | . . . 4 ⊢ (◡ GraphLocIso “ (V ∖ 1o)) ⊆ dom GraphLocIso | |
| 3 | grlimfn 47946 | . . . . 5 ⊢ GraphLocIso Fn (V × V) | |
| 4 | 3 | fndmi 6672 | . . . 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 5703 | . 2 ⊢ Rel (V × V) | |
| 8 | relss 5791 | . 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 3480 ∖ cdif 3948 ⊆ wss 3951 × cxp 5683 ◡ccnv 5684 dom cdm 5685 “ cima 5688 Rel wrel 5690 1oc1o 8499 GraphLocIso cgrlim 47943 ≃𝑙𝑔𝑟 cgrlic 47944 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-f1o 6568 df-fv 6569 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-grlim 47945 df-grlic 47948 |
| This theorem is referenced by: (None) |
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