Theorem grlicen 49330

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 49317	. 2 ⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 ↔ (𝑅 GraphLocIso 𝑆) ≠ ∅)
2		n0 4303	. . 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 49298	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GraphLocIso 𝑆) → 𝑓:𝐵⟶1-1onto𝐶)
6	3	fvexi 6925	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 9030	. . . . 5 ⊢ (𝑓:𝐵⟶1-1onto𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 18	. . . 4 ⊢ (𝑓 ∈ (𝑅 GraphLocIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1965	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GraphLocIso 𝑆) → 𝐵 ≈ 𝐶)
10	2, 9	sylbi 220	. 2 ⊢ ((𝑅 GraphLocIso 𝑆) ≠ ∅ → 𝐵 ≈ 𝐶)
11	1, 10	sylbi 220	1 ⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281   class class class wbr 5106  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446   ≈ cen 9001  Vtxcvtx 29607   GraphLocIso cgrlim 49289   ≃_𝑙𝑔𝑟 cgrlic 49290

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-1o 8507  df-en 9005  df-grlim 49291  df-grlic 49294

This theorem is used by: (None)