Theorem grlicen 47913

Description: Locally isomorphic graphs have equinumerous sets of vertices. (Contributed by AV, 11-Jun-2025.)

Hypotheses

Ref	Expression
grlicen.b	⊢ 𝐵 = (Vtx‘𝑅)
grlicen.c	⊢ 𝐶 = (Vtx‘𝑆)

Assertion

Ref	Expression
grlicen	⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)

Proof of Theorem grlicen

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgrlic 47900	. 2 ⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 ↔ (𝑅 GraphLocIso 𝑆) ≠ ∅)
2		n0 4359	. . 3 ⊢ ((𝑅 GraphLocIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GraphLocIso 𝑆))
3		grlicen.b	. . . . . 6 ⊢ 𝐵 = (Vtx‘𝑅)
4		grlicen.c	. . . . . 6 ⊢ 𝐶 = (Vtx‘𝑆)
5	3, 4	grlimf1o 47888	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GraphLocIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6921	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 9012	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 17	. . . 4 ⊢ (𝑓 ∈ (𝑅 GraphLocIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1928	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GraphLocIso 𝑆) → 𝐵 ≈ 𝐶)
10	2, 9	sylbi 217	. 2 ⊢ ((𝑅 GraphLocIso 𝑆) ≠ ∅ → 𝐵 ≈ 𝐶)
11	1, 10	sylbi 217	1 ⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)

Syntax hints:   → wi 4   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∅c0 4339   class class class wbr 5148  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431   ≈ cen 8981  Vtxcvtx 29028   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-rep 5285  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-reu 3379  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-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-en 8985  df-grlim 47881  df-grlic 47884

This theorem is referenced by: (None)