gricen - Mathbox for Alexander van der Vekens

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Theorem gricen 49162

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 49149	. 2 ⊢ (𝑅 ≃_𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
2		n0 4302	. . 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 49121	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6907	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 9010	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 18	. . . 4 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1964	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝐵 ≈ 𝐶)
10	2, 9	sylbi 220	. 2 ⊢ ((𝑅 GraphIso 𝑆) ≠ ∅ → 𝐵 ≈ 𝐶)
11	1, 10	sylbi 220	1 ⊢ (𝑅 ≃_𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∃wex 1811 ∈ wcel 2146 ≠ wne 2958 ∅c0 4280 class class class wbr 5105 –1-1-onto→wf1o 6545 ‘cfv 6546 (class class class)co 7427 ≈ cen 8981 Vtxcvtx 29586 GraphIso cgrim 49112 ≃_𝑔𝑟 cgric 49113

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-id 5549 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7430 df-oprab 7431 df-mpo 7432 df-1st 8008 df-2nd 8009 df-1o 8487 df-map 8860 df-en 8985 df-grim 49115 df-gric 49118

This theorem is used by: (None)

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