Theorem grlimgredgex 48498

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 2740	. . . 4 ⊢ (𝐺 ClNeighbVtx 𝐴) = (𝐺 ClNeighbVtx 𝐴)
8		grlimgredgex.i	. . . 4 ⊢ 𝐼 = (Edg‘𝐺)
9		eqid 2740	. . . 4 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}
10		eqid 2740	. . . 4 ⊢ (𝐻 ClNeighbVtx (𝐹‘𝐴)) = (𝐻 ClNeighbVtx (𝐹‘𝐴))
11		grlimgredgex.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐻)
12		eqid 2740	. . . 4 ⊢ {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} = {𝑥 ∈ 𝐸 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}
13	7, 8, 9, 10, 11, 12	grlimprclnbgrvtx 48497	. . 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 1397	. 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 6774	. . . . . . . . . . 11 ⊢ (𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) → 𝑓:(𝐺 ClNeighbVtx 𝐴)⟶(𝐻 ClNeighbVtx (𝐹‘𝐴)))
16	15	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝑓:(𝐺 ClNeighbVtx 𝐴)⟶(𝐻 ClNeighbVtx (𝐹‘𝐴)))
17		uspgrupgr 29272	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
18	1, 17	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
19	4, 5	jca 516	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
20	18, 19, 6	3jca 1134	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ∈ UPGraph ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ {𝐴, 𝐵} ∈ 𝐼))
21		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
22	21, 8	upgrpredgv 29233	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ {𝐴, 𝐵} ∈ 𝐼) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
23		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → 𝐵 ∈ (Vtx‘𝐺))
24	20, 22, 23	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (Vtx‘𝐺))
25		simpl 483	. . . . . . . . . . . . 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 48337	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ {𝐴, 𝐵} ∈ 𝐼) → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
28	24, 26, 6, 27	syl3anc 1379	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
29	28	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
30	16, 29	ffvelcdmd 7033	. . . . . . . . 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 48328	. . . . . . . . 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 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‘𝐵) ∈ 𝑉)
35		preq2 4673	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝐵) → {(𝐹‘𝐴), 𝑣} = {(𝐹‘𝐴), (𝑓‘𝐵)})
36	35	eleq1d 2825	. . . . . . . 8 ⊢ (𝑣 = (𝑓‘𝐵) → ({(𝐹‘𝐴), 𝑣} ∈ 𝐸 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐸))
37	36	adantl 482	. . . . . . 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 3947	. . . . . . . . . 10 ⊢ (𝑥 = {(𝐹‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴)) ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
39	38	elrab 3636	. . . . . . . . 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 482	. . . . . . 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 3569	. . . . . 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 413	. . . . 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 48327	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Vtx‘𝐺) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
45	26, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
46	45	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
47	16, 46	ffvelcdmd 7033	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
48	31	clnbgrisvtx 48328	. . . . . . . . 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 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) ∈ 𝑉)
51		preq2 4673	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝐴) → {(𝐹‘𝐴), 𝑣} = {(𝐹‘𝐴), (𝑓‘𝐴)})
52	51	eleq1d 2825	. . . . . . . 8 ⊢ (𝑣 = (𝑓‘𝐴) → ({(𝐹‘𝐴), 𝑣} ∈ 𝐸 ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐸))
53	52	adantl 482	. . . . . . 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 3947	. . . . . . . . . 10 ⊢ (𝑥 = {(𝐹‘𝐴), (𝑓‘𝐴)} → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴)) ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
55	54	elrab 3636	. . . . . . . . 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 482	. . . . . . 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 3569	. . . . . 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 413	. . . . 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 865	. . . 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 454	. . 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 1940	. 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 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃wrex 3064  {crab 3392   ⊆ wss 3890  {cpr 4564  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363  Vtxcvtx 29090  Edgcedg 29141  UPGraphcupgr 29174  USPGraphcuspgr 29242   ClNeighbVtx cclnbgr 48316   GraphLocIso cgrlim 48474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9823  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-n0 12436  df-xnn0 12509  df-z 12523  df-uz 12787  df-fz 13460  df-hash 14291  df-vtx 29092  df-iedg 29093  df-edg 29142  df-uhgr 29152  df-upgr 29176  df-uspgr 29244  df-nbgr 29427  df-clnbgr 48317  df-isubgr 48359  df-grim 48376  df-gric 48379  df-grlim 48476

This theorem is referenced by: (None)