Theorem grlicen 48013

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 48000	. 2 ⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 ↔ (𝑅 GraphLocIso 𝑆) ≠ ∅)
2		n0 4319	. . 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 47988	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GraphLocIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6875	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 8947	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 17	. . . 4 ⊢ (𝑓 ∈ (𝑅 GraphLocIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1930	. . 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 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∅c0 4299   class class class wbr 5110  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390   ≈ cen 8918  Vtxcvtx 28930   GraphLocIso cgrlim 47979   ≃_𝑙𝑔𝑟 cgrlic 47980

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-1o 8437  df-en 8922  df-grlim 47981  df-grlic 47984

This theorem is referenced by: (None)