Theorem isomuspgr 42552

Description: The isomorphy relation for two simple pseudographs. This corresponds to the definition in [Bollobas] p. 3. (Contributed by AV, 1-Dec-2022.)

Hypotheses

Ref	Expression
isomushgr.v	⊢ 𝑉 = (Vtx‘𝐴)
isomushgr.w	⊢ 𝑊 = (Vtx‘𝐵)
isomushgr.e	⊢ 𝐸 = (Edg‘𝐴)
isomushgr.k	⊢ 𝐾 = (Edg‘𝐵)

Assertion

Ref	Expression
isomuspgr	⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))))

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑎,𝑏,𝑓   𝐸,𝑎,𝑏   𝐾,𝑎,𝑏   𝑉,𝑎,𝑏   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏   𝑊,𝑎,𝑏

Allowed substitution hints:   𝐸(𝑓)   𝐾(𝑓)   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem isomuspgr

Dummy variables 𝑒 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrushgr 26474	. . 3 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ USHGraph)
2		uspgrushgr 26474	. . 3 ⊢ (𝐵 ∈ USPGraph → 𝐵 ∈ USHGraph)
3		isomushgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐴)
4		isomushgr.w	. . . 4 ⊢ 𝑊 = (Vtx‘𝐵)
5		isomushgr.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐴)
6		isomushgr.k	. . . 4 ⊢ 𝐾 = (Edg‘𝐵)
7	3, 4, 5, 6	isomushgr 42544	. . 3 ⊢ ((𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
8	1, 2, 7	syl2an 591	. 2 ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
9		imaeq2 5703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑓 “ 𝑒) = (𝑓 “ {𝑎, 𝑏}))
10		fveq2 6433	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑔‘𝑒) = (𝑔‘{𝑎, 𝑏}))
11	9, 10	eqeq12d 2840	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
12	11	rspccv 3523	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
13	12	adantl 475	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
14	13	imp 397	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏}))
15		f1ofn 6379	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓 Fn 𝑉)
16	15	ad3antlr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑓 Fn 𝑉)
17		simprl 789	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
18		simprr 791	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
19		fnimapr 6509	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑓 “ {𝑎, 𝑏}) = {(𝑓‘𝑎), (𝑓‘𝑏)})
20	16, 17, 18, 19	syl3anc 1496	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑓 “ {𝑎, 𝑏}) = {(𝑓‘𝑎), (𝑓‘𝑏)})
21	20	eqeq1d 2827	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏}) ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} = (𝑔‘{𝑎, 𝑏})))
22	21	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → ((𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏}) ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} = (𝑔‘{𝑎, 𝑏})))
23	22	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) ∧ {𝑎, 𝑏} ∈ 𝐸) → ((𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏}) ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} = (𝑔‘{𝑎, 𝑏})))
24		f1of 6378	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐸–1-1-onto→𝐾 → 𝑔:𝐸⟶𝐾)
25	24	ad3antlr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → 𝑔:𝐸⟶𝐾)
26	25	ffvelrnda 6608	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑔‘{𝑎, 𝑏}) ∈ 𝐾)
27		eleq1 2894	. . . . . . . . . . . . . . . 16 ⊢ ({(𝑓‘𝑎), (𝑓‘𝑏)} = (𝑔‘{𝑎, 𝑏}) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾 ↔ (𝑔‘{𝑎, 𝑏}) ∈ 𝐾))
28	26, 27	syl5ibrcom 239	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) ∧ {𝑎, 𝑏} ∈ 𝐸) → ({(𝑓‘𝑎), (𝑓‘𝑏)} = (𝑔‘{𝑎, 𝑏}) → {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))
29	23, 28	sylbid 232	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) ∧ {𝑎, 𝑏} ∈ 𝐸) → ((𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏}) → {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))
30	14, 29	mpd 15	. . . . . . . . . . . . 13 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) ∧ {𝑎, 𝑏} ∈ 𝐸) → {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)
31	30	exp41 427	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ({𝑎, 𝑏} ∈ 𝐸 → {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))))
32	31	com23 86	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ 𝑔:𝐸–1-1-onto→𝐾) → (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ({𝑎, 𝑏} ∈ 𝐸 → {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))))
33	32	impr 448	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ({𝑎, 𝑏} ∈ 𝐸 → {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)))
34	33	imp 397	. . . . . . . . 9 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑎, 𝑏} ∈ 𝐸 → {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))
35	3, 4, 5, 6	isomuspgrlem1 42545	. . . . . . . . 9 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾 → {𝑎, 𝑏} ∈ 𝐸))
36	34, 35	impbid 204	. . . . . . . 8 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))
37	36	ralrimivva 3180	. . . . . . 7 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))
38	37	ex 403	. . . . . 6 ⊢ (((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)))
39	38	exlimdv 2034	. . . . 5 ⊢ (((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) → (∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)))
40	3, 4, 5, 6	isomuspgrlem2 42551	. . . . 5 ⊢ (((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))))
41	39, 40	impbid 204	. . . 4 ⊢ (((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) → (∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)))
42	41	pm5.32da 576	. . 3 ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → ((𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))))
43	42	exbidv 2022	. 2 ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → (∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))))
44	8, 43	bitrd 271	1 ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658  ∃wex 1880   ∈ wcel 2166  ∀wral 3117  {cpr 4399   class class class wbr 4873   “ cima 5345   Fn wfn 6118  ⟶wf 6119  –1-1-onto→wf1o 6122  ‘cfv 6123  Vtxcvtx 26294  Edgcedg 26345  USHGraphcushgr 26355  USPGraphcuspgr 26447   IsomGr cisomgr 42537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-n0 11619  df-xnn0 11691  df-z 11705  df-uz 11969  df-fz 12620  df-hash 13411  df-edg 26346  df-uhgr 26356  df-ushgr 26357  df-upgr 26380  df-uspgr 26449  df-isomgr 42539

This theorem is referenced by: (None)