Theorem grlimgredgex 47985

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 2729	. . . 4 ⊢ (𝐺 ClNeighbVtx 𝐴) = (𝐺 ClNeighbVtx 𝐴)
8		grlimgredgex.i	. . . 4 ⊢ 𝐼 = (Edg‘𝐺)
9		eqid 2729	. . . 4 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}
10		eqid 2729	. . . 4 ⊢ (𝐻 ClNeighbVtx (𝐹‘𝐴)) = (𝐻 ClNeighbVtx (𝐹‘𝐴))
11		grlimgredgex.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐻)
12		eqid 2729	. . . 4 ⊢ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} = {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}
13	7, 8, 9, 10, 11, 12	grlimprclnbgrvtx 47984	. . 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 29141	. . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
22	21, 8	upgrpredgv 29102	. . . . . . . . . . . . 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 47824	. . . . . . . . . . . 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 7023	. . . . . . . . 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 47815	. . . . . . . . 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 4688	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝐵) → {(𝐹‘𝐴), 𝑣} = {(𝐹‘𝐴), (𝑓‘𝐵)})
36	35	eleq1d 2813	. . . . . . . 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 3963	. . . . . . . . . 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 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 3581	. . . . . 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 47814	. . . . . . . . . . . 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 7023	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
48	31	clnbgrisvtx 47815	. . . . . . . . 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 4688	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝐴) → {(𝐹‘𝐴), 𝑣} = {(𝐹‘𝐴), (𝑓‘𝐴)})
52	51	eleq1d 2813	. . . . . . . 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 3963	. . . . . . . . . 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 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 3581	. . . . . 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 1933	. 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 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  {crab 3396   ⊆ wss 3905  {cpr 4581  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Vtxcvtx 28959  Edgcedg 29010  UPGraphcupgr 29043  USPGraphcuspgr 29111   ClNeighbVtx cclnbgr 47803   GraphLocIso cgrlim 47961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  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 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-xnn0 12476  df-z 12490  df-uz 12754  df-fz 13429  df-hash 14256  df-vtx 28961  df-iedg 28962  df-edg 29011  df-uhgr 29021  df-upgr 29045  df-uspgr 29113  df-nbgr 29296  df-clnbgr 47804  df-isubgr 47846  df-grim 47863  df-gric 47866  df-grlim 47963

This theorem is referenced by: (None)