Theorem grlimgredgex 48586

Description: Local isomorphisms between simple pseudographs map an edge onto an edge with an endpoint being the image of one of the endpoints of the first edge under the local isomorphism. (Contributed by AV, 28-Dec-2025.)

Hypotheses

Ref	Expression
grlimgredgex.i	⊢ 𝐼 = (Edg‘𝐺)
grlimgredgex.e	⊢ 𝐸 = (Edg‘𝐻)
grlimgredgex.v	⊢ 𝑉 = (Vtx‘𝐻)
grlimgredgex.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
grlimgredgex.b	⊢ (𝜑 → 𝐵 ∈ 𝑌)
grlimgredgex.p	⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝐼)
grlimgredgex.g	⊢ (𝜑 → 𝐺 ∈ USPGraph)
grlimgredgex.h	⊢ (𝜑 → 𝐻 ∈ USPGraph)
grlimgredgex.f	⊢ (𝜑 → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))

Assertion

Ref	Expression
grlimgredgex	⊢ (𝜑 → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸)

Distinct variable groups:   𝑣,𝐴   𝑣,𝐵   𝑣,𝐸   𝑣,𝐹   𝑣,𝐺   𝑣,𝐻   𝑣,𝑉   𝜑,𝑣

Allowed substitution hints:   𝐼(𝑣)   𝑋(𝑣)   𝑌(𝑣)

Proof of Theorem grlimgredgex

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grlimgredgex.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ USPGraph)
2		grlimgredgex.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USPGraph)
3		grlimgredgex.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
4		grlimgredgex.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
5		grlimgredgex.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
6		grlimgredgex.p	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝐼)
7		eqid 2761	. . . 4 ⊢ (𝐺 ClNeighbVtx 𝐴) = (𝐺 ClNeighbVtx 𝐴)
8		grlimgredgex.i	. . . 4 ⊢ 𝐼 = (Edg‘𝐺)
9		eqid 2761	. . . 4 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}
10		eqid 2761	. . . 4 ⊢ (𝐻 ClNeighbVtx (𝐹‘𝐴)) = (𝐻 ClNeighbVtx (𝐹‘𝐴))
11		grlimgredgex.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐻)
12		eqid 2761	. . . 4 ⊢ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} = {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}
13	7, 8, 9, 10, 11, 12	grlimprclnbgrvtx 48585	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))})))
14	1, 2, 3, 4, 5, 6, 13	syl213anc 1407	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))})))
15		f1of 6802	. . . . . . . . . . 11 ⊢ (𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) → 𝑓:(𝐺 ClNeighbVtx 𝐴)⟶(𝐻 ClNeighbVtx (𝐹‘𝐴)))
16	15	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝑓:(𝐺 ClNeighbVtx 𝐴)⟶(𝐻 ClNeighbVtx (𝐹‘𝐴)))
17		uspgrupgr 29325	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
18	1, 17	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
19	4, 5	jca 519	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
20	18, 19, 6	3jca 1140	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ∈ UPGraph ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ {𝐴, 𝐵} ∈ 𝐼))
21		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
22	21, 8	upgrpredgv 29286	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ {𝐴, 𝐵} ∈ 𝐼) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
23		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → 𝐵 ∈ (Vtx‘𝐺))
24	20, 22, 23	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (Vtx‘𝐺))
25		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → 𝐴 ∈ (Vtx‘𝐺))
26	20, 22, 25	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (Vtx‘𝐺))
27	21, 8	predgclnbgrel 48425	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ {𝐴, 𝐵} ∈ 𝐼) → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
28	24, 26, 6, 27	syl3anc 1389	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
29	28	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
30	16, 29	ffvelcdmd 7062	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → (𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
31		grlimgredgex.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐻)
32	31	clnbgrisvtx 48416	. . . . . . . . 9 ⊢ ((𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)) → (𝑓‘𝐵) ∈ 𝑉)
33	30, 32	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → (𝑓‘𝐵) ∈ 𝑉)
34	33	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}) → (𝑓‘𝐵) ∈ 𝑉)
35		preq2 4692	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝐵) → {(𝐹‘𝐴), 𝑣} = {(𝐹‘𝐴), (𝑓‘𝐵)})
36	35	eleq1d 2846	. . . . . . . 8 ⊢ (𝑣 = (𝑓‘𝐵) → ({(𝐹‘𝐴), 𝑣} ∈ 𝐸 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐸))
37	36	adantl 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}) ∧ 𝑣 = (𝑓‘𝐵)) → ({(𝐹‘𝐴), 𝑣} ∈ 𝐸 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐸))
38		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑥 = {(𝐹‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴)) ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
39	38	elrab 3650	. . . . . . . . 9 ⊢ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ↔ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐸 ∧ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
40	39	simplbi 500	. . . . . . . 8 ⊢ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} → {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐸)
41	40	adantl 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}) → {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐸)
42	34, 37, 41	rspcedvd 3583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}) → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸)
43	42	ex 416	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸))
44	21	clnbgrvtxel 48415	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Vtx‘𝐺) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
45	26, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
46	45	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
47	16, 46	ffvelcdmd 7062	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
48	31	clnbgrisvtx 48416	. . . . . . . . 9 ⊢ ((𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)) → (𝑓‘𝐴) ∈ 𝑉)
49	47, 48	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → (𝑓‘𝐴) ∈ 𝑉)
50	49	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}) → (𝑓‘𝐴) ∈ 𝑉)
51		preq2 4692	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝐴) → {(𝐹‘𝐴), 𝑣} = {(𝐹‘𝐴), (𝑓‘𝐴)})
52	51	eleq1d 2846	. . . . . . . 8 ⊢ (𝑣 = (𝑓‘𝐴) → ({(𝐹‘𝐴), 𝑣} ∈ 𝐸 ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐸))
53	52	adantl 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}) ∧ 𝑣 = (𝑓‘𝐴)) → ({(𝐹‘𝐴), 𝑣} ∈ 𝐸 ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐸))
54		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑥 = {(𝐹‘𝐴), (𝑓‘𝐴)} → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴)) ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
55	54	elrab 3650	. . . . . . . . 9 ⊢ ({(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ↔ ({(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐸 ∧ {(𝐹‘𝐴), (𝑓‘𝐴)} ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
56	55	simplbi 500	. . . . . . . 8 ⊢ ({(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐸)
57	56	adantl 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐸)
58	50, 53, 57	rspcedvd 3583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}) → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸)
59	58	ex 416	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → ({(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸))
60	43, 59	jaod 870	. . . 4 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → (({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}) → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸))
61	60	expimpd 457	. . 3 ⊢ (𝜑 → ((𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))})) → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸))
62	61	exlimdv 1952	. 2 ⊢ (𝜑 → (∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))})) → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸))
63	14, 62	mpd 15	1 ⊢ (𝜑 → ∃𝑣 ∈ 𝑉 {(𝐹‘𝐴), 𝑣} ∈ 𝐸)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∃wrex 3085  {crab 3413   ⊆ wss 3904  {cpr 4583  ⟶wf 6513  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392  Vtxcvtx 29143  Edgcedg 29194  UPGraphcupgr 29227  USPGraphcuspgr 29295   ClNeighbVtx cclnbgr 48404   GraphLocIso cgrlim 48562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-dju 9856  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-n0 12479  df-xnn0 12552  df-z 12566  df-uz 12837  df-fz 13510  df-hash 14341  df-vtx 29145  df-iedg 29146  df-edg 29195  df-uhgr 29205  df-upgr 29229  df-uspgr 29297  df-nbgr 29480  df-clnbgr 48405  df-isubgr 48447  df-grim 48464  df-gric 48467  df-grlim 48564

This theorem is referenced by: (None)