Theorem cycldlenngric 48909

Description: Two simple pseudographs are not isomorphic if one has a cycle and the other has no cycle of the same length. (Contributed by AV, 6-Nov-2025.)

Assertion

Ref	Expression
cycldlenngric	⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ∧ ¬ ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ¬ 𝐺 ≃𝑔𝑟 𝐻))

Distinct variable groups:   𝑓,𝐺,𝑝   𝑓,𝐻,𝑝   𝑓,𝑁,𝑝

Proof of Theorem cycldlenngric

Dummy variables 𝑔 𝑖 𝑗 𝑞 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgric 48893	. . . . . . . 8 ⊢ (𝐺 ≃𝑔𝑟 𝐻 ↔ (𝐺 GraphIso 𝐻) ≠ ∅)
2		n0rex 4305	. . . . . . . . 9 ⊢ ((𝐺 GraphIso 𝐻) ≠ ∅ → ∃𝑖 ∈ (𝐺 GraphIso 𝐻)𝑖 ∈ (𝐺 GraphIso 𝐻))
3		eqid 2760	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2760	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5		simprll 791	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝐺 ∈ USPGraph)
6		simprlr 792	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝐻 ∈ USPGraph)
7		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝑖 ∈ (𝐺 GraphIso 𝐻))
8		2fveq3 6886	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑗 → ((iEdg‘𝐺)‘(𝑓‘𝑥)) = ((iEdg‘𝐺)‘(𝑓‘𝑗)))
9	8	imaeq2d 6058	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑗 → (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))) = (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗))))
10	9	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑗 → (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))) = (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
11	10	cbvmptv 5209	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) = (𝑗 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
12		cycliswlk 30297	. . . . . . . . . . . . . . 15 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
13	12	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → 𝑓(Walks‘𝐺)𝑝)
14	13	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝑓(Walks‘𝐺)𝑝)
15	3, 4, 5, 6, 7, 11, 14	upgrimwlklen 48884	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)))
16		simprrl 793	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝑓(Cycles‘𝐺)𝑝)
17	3, 4, 5, 6, 7, 11, 16	upgrimcycls 48892	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝))
18		simp3 1156	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝))
19		simp2r 1219	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓))
20		simprrr 794	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → (♯‘𝑓) = 𝑁)
21	20	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → (♯‘𝑓) = 𝑁)
22	19, 21	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁)
23		vex 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑖 ∈ V
24		vex 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑝 ∈ V
25	23, 24	coex 7933	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∘ 𝑝) ∈ V
26		vex 3454	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
27	26	dmex 7912	. . . . . . . . . . . . . . 15 ⊢ dom 𝑓 ∈ V
28	27	mptex 7225	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) ∈ V
29		breq12 5108	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) ∧ 𝑞 = (𝑖 ∘ 𝑝)) → (𝑔(Cycles‘𝐻)𝑞 ↔ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)))
30	29	ancoms 464	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑖 ∘ 𝑝) ∧ 𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) → (𝑔(Cycles‘𝐻)𝑞 ↔ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)))
31		fveqeq2 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) → ((♯‘𝑔) = 𝑁 ↔ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁))
32	31	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑖 ∘ 𝑝) ∧ 𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) → ((♯‘𝑔) = 𝑁 ↔ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁))
33	30, 32	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ ((𝑞 = (𝑖 ∘ 𝑝) ∧ 𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) → ((𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁) ↔ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁)))
34	25, 28, 33	spc2ev 3561	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
35	18, 22, 34	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
36	15, 17, 35	mpd3an23 1492	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
37	36	ex 418	. . . . . . . . . 10 ⊢ (𝑖 ∈ (𝐺 GraphIso 𝐻) → (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
38	37	rexlimivw 3159	. . . . . . . . 9 ⊢ (∃𝑖 ∈ (𝐺 GraphIso 𝐻)𝑖 ∈ (𝐺 GraphIso 𝐻) → (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
39	2, 38	syl 18	. . . . . . . 8 ⊢ ((𝐺 GraphIso 𝐻) ≠ ∅ → (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
40	1, 39	sylbi 220	. . . . . . 7 ⊢ (𝐺 ≃𝑔𝑟 𝐻 → (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
41	40	expdcom 420	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → (𝐺 ≃𝑔𝑟 𝐻 → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))))
42	41	exlimdvv 1967	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → (𝐺 ≃𝑔𝑟 𝐻 → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))))
43	42	imp 412	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ ∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → (𝐺 ≃𝑔𝑟 𝐻 → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
44		breq12 5108	. . . . . . 7 ⊢ ((𝑓 = 𝑔 ∧ 𝑝 = 𝑞) → (𝑓(Cycles‘𝐻)𝑝 ↔ 𝑔(Cycles‘𝐻)𝑞))
45	44	ancoms 464	. . . . . 6 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑓(Cycles‘𝐻)𝑝 ↔ 𝑔(Cycles‘𝐻)𝑞))
46		fveqeq2 6890	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
47	46	adantl 487	. . . . . 6 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
48	45, 47	anbi12d 644	. . . . 5 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ (𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
49	48	cbvex2vw 2074	. . . 4 ⊢ (∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
50	43, 49	imbitrrdi 255	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ ∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → (𝐺 ≃𝑔𝑟 𝐻 → ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁)))
51	50	con3d 153	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ ∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → (¬ ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) → ¬ 𝐺 ≃𝑔𝑟 𝐻))
52	51	expimpd 459	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ∧ ¬ ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ¬ 𝐺 ≃𝑔𝑟 𝐻))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  ∅c0 4279   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5654  dom cdm 5655   “ cima 5658   ∘ ccom 5659  ‘cfv 6535  (class class class)co 7416  ♯chash 14419  iEdgciedg 29486  USPGraphcuspgr 29640  Walkscwlks 30088  Cyclesccycls 30283   GraphIso cgrim 48856   ≃_𝑔𝑟 cgric 48857

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8703  df-map 8835  df-pm 8836  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9931  df-card 9969  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-nn 12283  df-2 12352  df-n0 12554  df-xnn0 12627  df-z 12641  df-uz 12913  df-fz 13587  df-fzo 13735  df-hash 14420  df-word 14604  df-edg 29537  df-uhgr 29547  df-upgr 29571  df-uspgr 29642  df-wlks 30091  df-trls 30186  df-pths 30210  df-cycls 30285  df-grim 48859  df-gric 48862

This theorem is used by:  gpg5ngric  49109