gricen - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > gricen		Structured version Visualization version GIF version

Theorem gricen 49022

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 49009	. 2 ⊢ (𝑅 ≃_𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
2		n0 4300	. . 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 48981	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6899	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 8999	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 18	. . . 4 ⊢ (𝑓 ∈ (𝑅 GraphIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1963	. . 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 1570 ∃wex 1812 ∈ wcel 2145 ≠ wne 2956 ∅c0 4279 class class class wbr 5103 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ≈ cen 8970 Vtxcvtx 29574 GraphIso cgrim 48972 ≃_𝑔𝑟 cgric 48973

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-1o 8476 df-map 8849 df-en 8974 df-grim 48975 df-gric 48978

This theorem is used by: (None)

W3C validator