Theorem isomuspgr 45704

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 27833	. . 3 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ USHGraph)
2		uspgrushgr 27833	. . 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 45696	. . 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 597	. 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 5999	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑓 “ 𝑒) = (𝑓 “ {𝑎, 𝑏}))
10		fveq2 6829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑔‘𝑒) = (𝑔‘{𝑎, 𝑏}))
11	9, 10	eqeq12d 2753	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
12	11	rspccv 3570	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
13	12	adantl 483	. . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . 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 6772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓 Fn 𝑉)
16	15	ad3antlr 729	. . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . 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 6912	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑓 “ {𝑎, 𝑏}) = {(𝑓‘𝑎), (𝑓‘𝑏)})
20	16, 17, 18, 19	syl3anc 1371	. . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . 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 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}) ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} = (𝑔‘{𝑎, 𝑏})))
24		f1of 6771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔:𝐸–1-1-onto→𝐾 → 𝑔:𝐸⟶𝐾)
25	24	ad3antlr 729	. . . . . . . . . . . . . . . . 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	ffvelcdmda 7021	. . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . 16 ⊢ ({(𝑓‘𝑎), (𝑓‘𝑏)} = (𝑔‘{𝑎, 𝑏}) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾 ↔ (𝑔‘{𝑎, 𝑏}) ∈ 𝐾))
28	26, 27	syl5ibrcom 247	. . . . . . . . . . . . . . 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 436	. . . . . . . . . . . 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 456	. . . . . . . . . 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 408	. . . . . . . . 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 45697	. . . . . . . . 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 3194	. . . . . . 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 414	. . . . . 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 45703	. . . . 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 580	. . 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 279	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 397   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3062  {cpr 4579   class class class wbr 5096   “ cima 5627   Fn wfn 6478  ⟶wf 6479  –1-1-onto→wf1o 6482  ‘cfv 6483  Vtxcvtx 27654  Edgcedg 27705  USHGraphcushgr 27715  USPGraphcuspgr 27806   IsomGr cisomgr 45689

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-int 4899  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-1st 7903  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-1o 8371  df-2o 8372  df-oadd 8375  df-er 8573  df-en 8809  df-dom 8810  df-sdom 8811  df-fin 8812  df-dju 9762  df-card 9800  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-nn 12079  df-2 12141  df-n0 12339  df-xnn0 12411  df-z 12425  df-uz 12688  df-fz 13345  df-hash 14150  df-edg 27706  df-uhgr 27716  df-ushgr 27717  df-upgr 27740  df-uspgr 27808  df-isomgr 45691

This theorem is referenced by: (None)