Theorem isomuspgr 46581

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 28473	. . 3 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ USHGraph)
2		uspgrushgr 28473	. . 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 46573	. . 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 6055	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑓 “ 𝑒) = (𝑓 “ {𝑎, 𝑏}))
10		fveq2 6891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑔‘𝑒) = (𝑔‘{𝑎, 𝑏}))
11	9, 10	eqeq12d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ {𝑎, 𝑏}) = (𝑔‘{𝑎, 𝑏})))
12	11	rspccv 3609	. . . . . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . . . 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 6975	. . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . 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 6833	. . . . . . . . . . . . . . . . . 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 7086	. . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . 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 46574	. . . . . . . . 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 3200	. . . . . . 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 46580	. . . . 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 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3061  {cpr 4630   class class class wbr 5148   “ cima 5679   Fn wfn 6538  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  Vtxcvtx 28294  Edgcedg 28345  USHGraphcushgr 28355  USPGraphcuspgr 28446   IsomGr cisomgr 46566

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 397  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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-fz 13487  df-hash 14293  df-edg 28346  df-uhgr 28356  df-ushgr 28357  df-upgr 28380  df-uspgr 28448  df-isomgr 46568

This theorem is referenced by: (None)