gricen - Mathbox for Alexander van der Vekens

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Theorem gricen 48113

Description: Isomorphic graphs have equinumerous sets of vertices. (Contributed by AV, 3-May-2025.)

**Hypotheses**
Ref	Expression
gricen.b	⊢ 𝐵 = (Vtx‘𝑅)
gricen.c	⊢ 𝐶 = (Vtx‘𝑆)

**Assertion**
Ref	Expression
gricen	⊢ (𝑅 ≃_𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)

Proof of Theorem gricen Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgric 48100	. 2 ⊢ (𝑅 ≃_𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
2		n0 4303	. . 3 ⊢ ((𝑅 GraphIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GraphIso 𝑆))
3		gricen.b	. . . . . 6 ⊢ 𝐵 = (Vtx‘𝑅)
4		gricen.c	. . . . . 6 ⊢ 𝐶 = (Vtx‘𝑆)
5	3, 4	grimf1o 48072	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6846	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 8907	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 17	. . . 4 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1931	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝐵 ≈ 𝐶)
10	2, 9	sylbi 217	. 2 ⊢ ((𝑅 GraphIso 𝑆) ≠ ∅ → 𝐵 ≈ 𝐶)
11	1, 10	sylbi 217	1 ⊢ (𝑅 ≃_𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ≠ wne 2930 ∅c0 4283 class class class wbr 5096 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7356 ≈ cen 8878 Vtxcvtx 29018 GraphIso cgrim 48063 ≃_𝑔𝑟 cgric 48064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-1o 8395 df-map 8763 df-en 8882 df-grim 48066 df-gric 48069

This theorem is referenced by: (None)

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