Theorem grlicrel 47991

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 47966	. . 3 ⊢ ≃𝑙𝑔𝑟 = (◡ GraphLocIso “ (V ∖ 1o))
2		cnvimass 6037	. . . 4 ⊢ (◡ GraphLocIso “ (V ∖ 1o)) ⊆ dom GraphLocIso
3		grlimfn 47964	. . . . 5 ⊢ GraphLocIso Fn (V × V)
4	3	fndmi 6590	. . . 4 ⊢ dom GraphLocIso = (V × V)
5	2, 4	sseqtri 3986	. . 3 ⊢ (◡ GraphLocIso “ (V ∖ 1o)) ⊆ (V × V)
6	1, 5	eqsstri 3984	. 2 ⊢ ≃𝑙𝑔𝑟 ⊆ (V × V)
7		relxp 5641	. 2 ⊢ Rel (V × V)
8		relss 5729	. 2 ⊢ ( ≃𝑙𝑔𝑟 ⊆ (V × V) → (Rel (V × V) → Rel ≃𝑙𝑔𝑟 ))
9	6, 7, 8	mp2 9	1 ⊢ Rel ≃𝑙𝑔𝑟

Syntax hints:  Vcvv 3438   ∖ cdif 3902   ⊆ wss 3905   × cxp 5621  ^◡ccnv 5622  dom cdm 5623   “ cima 5626  Rel wrel 5628  1_oc1o 8388   GraphLocIso cgrlim 47961   ≃_𝑙𝑔𝑟 cgrlic 47962

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-f1o 6493  df-fv 6494  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-grlim 47963  df-grlic 47966

This theorem is referenced by: (None)