Theorem isomuspgr 44352

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 26968	. . 3 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ USHGraph)
2		uspgrushgr 26968	. . 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 44344	. . 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 598	. 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 5892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑓 “ 𝑒) = (𝑓 “ {𝑎, 𝑏}))
10		fveq2 6645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑔‘𝑒) = (𝑔‘{𝑎, 𝑏}))
11	9, 10	eqeq12d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
12	11	rspccv 3568	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
13	12	adantl 485	. . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . 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 6591	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓 Fn 𝑉)
16	15	ad3antlr 730	. . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . 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 6722	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑓 “ {𝑎, 𝑏}) = {(𝑓‘𝑎), (𝑓‘𝑏)})
20	16, 17, 18, 19	syl3anc 1368	. . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 6590	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐸–1-1-onto→𝐾 → 𝑔:𝐸⟶𝐾)
25	24	ad3antlr 730	. . . . . . . . . . . . . . . . 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 6828	. . . . . . . . . . . . . . . 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 2877	. . . . . . . . . . . . . . . 16 ⊢ ({(𝑓‘𝑎), (𝑓‘𝑏)} = (𝑔‘{𝑎, 𝑏}) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾 ↔ (𝑔‘{𝑎, 𝑏}) ∈ 𝐾))
28	26, 27	syl5ibrcom 250	. . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . 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 438	. . . . . . . . . . . 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 458	. . . . . . . . . 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 410	. . . . . . . . 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 44345	. . . . . . . . 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 215	. . . . . . . 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 3156	. . . . . . 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 416	. . . . . 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 1934	. . . . 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 44351	. . . . 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 215	. . . 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 582	. . 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 1922	. 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 282	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 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  {cpr 4527   class class class wbr 5030   “ cima 5522   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  Vtxcvtx 26789  Edgcedg 26840  USHGraphcushgr 26850  USPGraphcuspgr 26941   IsomGr cisomgr 44337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-hash 13687  df-edg 26841  df-uhgr 26851  df-ushgr 26852  df-upgr 26875  df-uspgr 26943  df-isomgr 44339

This theorem is referenced by: (None)