Theorem grlicrel 48482

Description: The "is locally isomorphic to" relation for graphs is a relation. (Contributed by AV, 9-Jun-2025.)

Assertion

Ref	Expression
grlicrel	⊢ Rel ≃𝑙𝑔𝑟

Proof of Theorem grlicrel

Step	Hyp	Ref	Expression
1		df-grlic 48457	. . 3 ⊢ ≃𝑙𝑔𝑟 = (◡ GraphLocIso “ (V ∖ 1o))
2		cnvimass 6047	. . . 4 ⊢ (◡ GraphLocIso “ (V ∖ 1o)) ⊆ dom GraphLocIso
3		grlimfn 48455	. . . . 5 ⊢ GraphLocIso Fn (V × V)
4	3	fndmi 6602	. . . 4 ⊢ dom GraphLocIso = (V × V)
5	2, 4	sseqtri 3970	. . 3 ⊢ (◡ GraphLocIso “ (V ∖ 1o)) ⊆ (V × V)
6	1, 5	eqsstri 3968	. 2 ⊢ ≃𝑙𝑔𝑟 ⊆ (V × V)
7		relxp 5649	. 2 ⊢ Rel (V × V)
8		relss 5738	. 2 ⊢ ( ≃𝑙𝑔𝑟 ⊆ (V × V) → (Rel (V × V) → Rel ≃𝑙𝑔𝑟 ))
9	6, 7, 8	mp2 9	1 ⊢ Rel ≃𝑙𝑔𝑟

Syntax hints:  Vcvv 3429   ∖ cdif 3886   ⊆ wss 3889   × cxp 5629  ^◡ccnv 5630  dom cdm 5631   “ cima 5634  Rel wrel 5636  1_oc1o 8398   GraphLocIso cgrlim 48452   ≃_𝑙𝑔𝑟 cgrlic 48453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-f1o 6505  df-fv 6506  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-grlim 48454  df-grlic 48457

This theorem is referenced by: (None)