gricen - Mathbox for Alexander van der Vekens

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Theorem gricen 47905

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 47892	. 2 ⊢ (𝑅 ≃_𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
2		n0 4333	. . 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 47864	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6895	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 8992	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 17	. . . 4 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1930	. . 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 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2933 ∅c0 4313 class class class wbr 5124 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 ≈ cen 8961 Vtxcvtx 28980 GraphIso cgrim 47855 ≃_𝑔𝑟 cgric 47856

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7993 df-2nd 7994 df-1o 8485 df-map 8847 df-en 8965 df-grim 47858 df-gric 47861

This theorem is referenced by: (None)

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