Theorem grlimgredgex 48124

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 2733	. . . 4 ⊢ (𝐺 ClNeighbVtx 𝐴) = (𝐺 ClNeighbVtx 𝐴)
8		grlimgredgex.i	. . . 4 ⊢ 𝐼 = (Edg‘𝐺)
9		eqid 2733	. . . 4 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}
10		eqid 2733	. . . 4 ⊢ (𝐻 ClNeighbVtx (𝐹‘𝐴)) = (𝐻 ClNeighbVtx (𝐹‘𝐴))
11		grlimgredgex.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐻)
12		eqid 2733	. . . 4 ⊢ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} = {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}
13	7, 8, 9, 10, 11, 12	grlimprclnbgrvtx 48123	. . 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 1391	. 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 6768	. . . . . . . . . . 11 ⊢ (𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) → 𝑓:(𝐺 ClNeighbVtx 𝐴)⟶(𝐻 ClNeighbVtx (𝐹‘𝐴)))
16	15	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝑓:(𝐺 ClNeighbVtx 𝐴)⟶(𝐻 ClNeighbVtx (𝐹‘𝐴)))
17		uspgrupgr 29158	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
18	1, 17	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
19	4, 5	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
20	18, 19, 6	3jca 1128	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ∈ UPGraph ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ {𝐴, 𝐵} ∈ 𝐼))
21		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
22	21, 8	upgrpredgv 29119	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ {𝐴, 𝐵} ∈ 𝐼) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
23		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → 𝐵 ∈ (Vtx‘𝐺))
24	20, 22, 23	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (Vtx‘𝐺))
25		simpl 482	. . . . . . . . . . . . 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 47963	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ {𝐴, 𝐵} ∈ 𝐼) → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
28	24, 26, 6, 27	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
29	28	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
30	16, 29	ffvelcdmd 7024	. . . . . . . . 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 47954	. . . . . . . . 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 480	. . . . . . 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 4686	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝐵) → {(𝐹‘𝐴), 𝑣} = {(𝐹‘𝐴), (𝑓‘𝐵)})
36	35	eleq1d 2818	. . . . . . . 8 ⊢ (𝑣 = (𝑓‘𝐵) → ({(𝐹‘𝐴), 𝑣} ∈ 𝐸 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐸))
37	36	adantl 481	. . . . . . 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 3956	. . . . . . . . . 10 ⊢ (𝑥 = {(𝐹‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴)) ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
39	38	elrab 3643	. . . . . . . . 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 497	. . . . . . . 8 ⊢ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} → {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐸)
41	40	adantl 481	. . . . . . 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 3575	. . . . . 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 412	. . . . 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 47953	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Vtx‘𝐺) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
45	26, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
46	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
47	16, 46	ffvelcdmd 7024	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
48	31	clnbgrisvtx 47954	. . . . . . . . 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 480	. . . . . . 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 4686	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝐴) → {(𝐹‘𝐴), 𝑣} = {(𝐹‘𝐴), (𝑓‘𝐴)})
52	51	eleq1d 2818	. . . . . . . 8 ⊢ (𝑣 = (𝑓‘𝐴) → ({(𝐹‘𝐴), 𝑣} ∈ 𝐸 ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐸))
53	52	adantl 481	. . . . . . 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 3956	. . . . . . . . . 10 ⊢ (𝑥 = {(𝐹‘𝐴), (𝑓‘𝐴)} → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴)) ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
55	54	elrab 3643	. . . . . . . . 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 497	. . . . . . . 8 ⊢ ({(𝐹‘𝐴), (𝑓‘𝐴)} ∈ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐸)
57	56	adantl 481	. . . . . . 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 3575	. . . . . 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 412	. . . . 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 859	. . . 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 453	. . 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 1934	. 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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3057  {crab 3396   ⊆ wss 3898  {cpr 4577  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7352  Vtxcvtx 28976  Edgcedg 29027  UPGraphcupgr 29060  USPGraphcuspgr 29128   ClNeighbVtx cclnbgr 47942   GraphLocIso cgrlim 48100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-oadd 8395  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-dju 9801  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-n0 12389  df-xnn0 12462  df-z 12476  df-uz 12739  df-fz 13410  df-hash 14240  df-vtx 28978  df-iedg 28979  df-edg 29028  df-uhgr 29038  df-upgr 29062  df-uspgr 29130  df-nbgr 29313  df-clnbgr 47943  df-isubgr 47985  df-grim 48002  df-gric 48005  df-grlim 48102

This theorem is referenced by: (None)