Theorem grlicrel 48011

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 47993	. . 3 ⊢ ≃𝑙𝑔𝑟 = (◡ GraphLocIso “ (V ∖ 1o))
2		cnvimass 6069	. . . 4 ⊢ (◡ GraphLocIso “ (V ∖ 1o)) ⊆ dom GraphLocIso
3		grlimfn 47991	. . . . 5 ⊢ GraphLocIso Fn (V × V)
4	3	fndmi 6642	. . . 4 ⊢ dom GraphLocIso = (V × V)
5	2, 4	sseqtri 4007	. . 3 ⊢ (◡ GraphLocIso “ (V ∖ 1o)) ⊆ (V × V)
6	1, 5	eqsstri 4005	. 2 ⊢ ≃𝑙𝑔𝑟 ⊆ (V × V)
7		relxp 5672	. 2 ⊢ Rel (V × V)
8		relss 5760	. 2 ⊢ ( ≃𝑙𝑔𝑟 ⊆ (V × V) → (Rel (V × V) → Rel ≃𝑙𝑔𝑟 ))
9	6, 7, 8	mp2 9	1 ⊢ Rel ≃𝑙𝑔𝑟

Syntax hints:  Vcvv 3459   ∖ cdif 3923   ⊆ wss 3926   × cxp 5652  ^◡ccnv 5653  dom cdm 5654   “ cima 5657  Rel wrel 5659  1_oc1o 8473   GraphLocIso cgrlim 47988   ≃_𝑙𝑔𝑟 cgrlic 47989

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-f1o 6538  df-fv 6539  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-grlim 47990  df-grlic 47993

This theorem is referenced by: (None)