gricen - Mathbox for Alexander van der Vekens

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Theorem gricen 48424

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 48411	. 2 ⊢ (𝑅 ≃_𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
2		n0 4282	. . 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 48383	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6842	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 8910	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 17	. . . 4 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1937	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝐵 ≈ 𝐶)
10	2, 9	sylbi 218	. 2 ⊢ ((𝑅 GraphIso 𝑆) ≠ ∅ → 𝐵 ≈ 𝐶)
11	1, 10	sylbi 218	1 ⊢ (𝑅 ≃_𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ≠ wne 2934 ∅c0 4262 class class class wbr 5073 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7357 ≈ cen 8881 Vtxcvtx 29084 GraphIso cgrim 48374 ≃_𝑔𝑟 cgric 48375

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7932 df-2nd 7933 df-1o 8396 df-map 8766 df-en 8885 df-grim 48377 df-gric 48380

This theorem is referenced by: (None)

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