Theorem isomuspgr 45286

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 27545	. . 3 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ USHGraph)
2		uspgrushgr 27545	. . 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 45278	. . 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 596	. 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 5965	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑓 “ 𝑒) = (𝑓 “ {𝑎, 𝑏}))
10		fveq2 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑔‘𝑒) = (𝑔‘{𝑎, 𝑏}))
11	9, 10	eqeq12d 2754	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
12	11	rspccv 3558	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
13	12	adantl 482	. . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . 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 6717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓 Fn 𝑉)
16	15	ad3antlr 728	. . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . 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 6852	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑓 “ {𝑎, 𝑏}) = {(𝑓‘𝑎), (𝑓‘𝑏)})
20	16, 17, 18, 19	syl3anc 1370	. . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 6716	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐸–1-1-onto→𝐾 → 𝑔:𝐸⟶𝐾)
25	24	ad3antlr 728	. . . . . . . . . . . . . . . . 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 6961	. . . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . 16 ⊢ ({(𝑓‘𝑎), (𝑓‘𝑏)} = (𝑔‘{𝑎, 𝑏}) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾 ↔ (𝑔‘{𝑎, 𝑏}) ∈ 𝐾))
28	26, 27	syl5ibrcom 246	. . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . 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 435	. . . . . . . . . . . 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 455	. . . . . . . . . 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 407	. . . . . . . . 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 45279	. . . . . . . . 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 211	. . . . . . . 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 3123	. . . . . . 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 413	. . . . . 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 1936	. . . . 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 45285	. . . . 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 211	. . . 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 579	. . 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 1924	. 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 278	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 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  {cpr 4563   class class class wbr 5074   “ cima 5592   Fn wfn 6428  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  Vtxcvtx 27366  Edgcedg 27417  USHGraphcushgr 27427  USPGraphcuspgr 27518   IsomGr cisomgr 45271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-edg 27418  df-uhgr 27428  df-ushgr 27429  df-upgr 27452  df-uspgr 27520  df-isomgr 45273

This theorem is referenced by: (None)