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Theorem cusgrsize 29324
Description: The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ β„•0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 βˆ’ 1)βˆ—π‘›) / 2), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)
Hypotheses
Ref Expression
cusgrsizeindb0.v 𝑉 = (Vtxβ€˜πΊ)
cusgrsizeindb0.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
cusgrsize ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) β†’ (β™―β€˜πΈ) = ((β™―β€˜π‘‰)C2))

Proof of Theorem cusgrsize
Dummy variables 𝑒 𝑓 𝑛 𝑣 𝑐 𝑀 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cusgrsizeindb0.e . . . . 5 𝐸 = (Edgβ€˜πΊ)
2 edgval 28918 . . . . 5 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
31, 2eqtri 2753 . . . 4 𝐸 = ran (iEdgβ€˜πΊ)
43a1i 11 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) β†’ 𝐸 = ran (iEdgβ€˜πΊ))
54fveq2d 6898 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) β†’ (β™―β€˜πΈ) = (β™―β€˜ran (iEdgβ€˜πΊ)))
6 cusgrsizeindb0.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
76opeq1i 4877 . . . 4 βŸ¨π‘‰, (iEdgβ€˜πΊ)⟩ = ⟨(Vtxβ€˜πΊ), (iEdgβ€˜πΊ)⟩
8 cusgrop 29307 . . . 4 (𝐺 ∈ ComplUSGraph β†’ ⟨(Vtxβ€˜πΊ), (iEdgβ€˜πΊ)⟩ ∈ ComplUSGraph)
97, 8eqeltrid 2829 . . 3 (𝐺 ∈ ComplUSGraph β†’ βŸ¨π‘‰, (iEdgβ€˜πΊ)⟩ ∈ ComplUSGraph)
10 fvex 6907 . . . 4 (iEdgβ€˜πΊ) ∈ V
11 fvex 6907 . . . . 5 (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∈ V
12 rabexg 5333 . . . . . 6 ((Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∈ V β†’ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐} ∈ V)
1312resiexd 7226 . . . . 5 ((Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∈ V β†’ ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}) ∈ V)
1411, 13ax-mp 5 . . . 4 ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}) ∈ V
15 rneq 5937 . . . . . 6 (𝑒 = (iEdgβ€˜πΊ) β†’ ran 𝑒 = ran (iEdgβ€˜πΊ))
1615fveq2d 6898 . . . . 5 (𝑒 = (iEdgβ€˜πΊ) β†’ (β™―β€˜ran 𝑒) = (β™―β€˜ran (iEdgβ€˜πΊ)))
17 fveq2 6894 . . . . . 6 (𝑣 = 𝑉 β†’ (β™―β€˜π‘£) = (β™―β€˜π‘‰))
1817oveq1d 7432 . . . . 5 (𝑣 = 𝑉 β†’ ((β™―β€˜π‘£)C2) = ((β™―β€˜π‘‰)C2))
1916, 18eqeqan12rd 2740 . . . 4 ((𝑣 = 𝑉 ∧ 𝑒 = (iEdgβ€˜πΊ)) β†’ ((β™―β€˜ran 𝑒) = ((β™―β€˜π‘£)C2) ↔ (β™―β€˜ran (iEdgβ€˜πΊ)) = ((β™―β€˜π‘‰)C2)))
20 rneq 5937 . . . . . 6 (𝑒 = 𝑓 β†’ ran 𝑒 = ran 𝑓)
2120fveq2d 6898 . . . . 5 (𝑒 = 𝑓 β†’ (β™―β€˜ran 𝑒) = (β™―β€˜ran 𝑓))
22 fveq2 6894 . . . . . 6 (𝑣 = 𝑀 β†’ (β™―β€˜π‘£) = (β™―β€˜π‘€))
2322oveq1d 7432 . . . . 5 (𝑣 = 𝑀 β†’ ((β™―β€˜π‘£)C2) = ((β™―β€˜π‘€)C2))
2421, 23eqeqan12rd 2740 . . . 4 ((𝑣 = 𝑀 ∧ 𝑒 = 𝑓) β†’ ((β™―β€˜ran 𝑒) = ((β™―β€˜π‘£)C2) ↔ (β™―β€˜ran 𝑓) = ((β™―β€˜π‘€)C2)))
25 vex 3467 . . . . . . 7 𝑣 ∈ V
26 vex 3467 . . . . . . 7 𝑒 ∈ V
2725, 26opvtxfvi 28878 . . . . . 6 (Vtxβ€˜βŸ¨π‘£, π‘’βŸ©) = 𝑣
2827eqcomi 2734 . . . . 5 𝑣 = (Vtxβ€˜βŸ¨π‘£, π‘’βŸ©)
29 eqid 2725 . . . . 5 (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) = (Edgβ€˜βŸ¨π‘£, π‘’βŸ©)
30 eqid 2725 . . . . 5 {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐} = {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}
31 eqid 2725 . . . . 5 ⟨(𝑣 βˆ– {𝑛}), ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})⟩ = ⟨(𝑣 βˆ– {𝑛}), ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})⟩
3228, 29, 30, 31cusgrres 29318 . . . 4 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ 𝑛 ∈ 𝑣) β†’ ⟨(𝑣 βˆ– {𝑛}), ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})⟩ ∈ ComplUSGraph)
33 rneq 5937 . . . . . . 7 (𝑓 = ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}) β†’ ran 𝑓 = ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}))
3433fveq2d 6898 . . . . . 6 (𝑓 = ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}) β†’ (β™―β€˜ran 𝑓) = (β™―β€˜ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})))
3534adantl 480 . . . . 5 ((𝑀 = (𝑣 βˆ– {𝑛}) ∧ 𝑓 = ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) β†’ (β™―β€˜ran 𝑓) = (β™―β€˜ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})))
36 fveq2 6894 . . . . . . 7 (𝑀 = (𝑣 βˆ– {𝑛}) β†’ (β™―β€˜π‘€) = (β™―β€˜(𝑣 βˆ– {𝑛})))
3736adantr 479 . . . . . 6 ((𝑀 = (𝑣 βˆ– {𝑛}) ∧ 𝑓 = ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) β†’ (β™―β€˜π‘€) = (β™―β€˜(𝑣 βˆ– {𝑛})))
3837oveq1d 7432 . . . . 5 ((𝑀 = (𝑣 βˆ– {𝑛}) ∧ 𝑓 = ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) β†’ ((β™―β€˜π‘€)C2) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2))
3935, 38eqeq12d 2741 . . . 4 ((𝑀 = (𝑣 βˆ– {𝑛}) ∧ 𝑓 = ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) β†’ ((β™―β€˜ran 𝑓) = ((β™―β€˜π‘€)C2) ↔ (β™―β€˜ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2)))
40 edgopval 28920 . . . . . . . . 9 ((𝑣 ∈ V ∧ 𝑒 ∈ V) β†’ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) = ran 𝑒)
4140el2v 3471 . . . . . . . 8 (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) = ran 𝑒
4241a1i 11 . . . . . . 7 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = 0) β†’ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) = ran 𝑒)
4342eqcomd 2731 . . . . . 6 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = 0) β†’ ran 𝑒 = (Edgβ€˜βŸ¨π‘£, π‘’βŸ©))
4443fveq2d 6898 . . . . 5 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = 0) β†’ (β™―β€˜ran 𝑒) = (β™―β€˜(Edgβ€˜βŸ¨π‘£, π‘’βŸ©)))
45 cusgrusgr 29288 . . . . . . 7 (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph β†’ βŸ¨π‘£, π‘’βŸ© ∈ USGraph)
46 usgruhgr 29055 . . . . . . 7 (βŸ¨π‘£, π‘’βŸ© ∈ USGraph β†’ βŸ¨π‘£, π‘’βŸ© ∈ UHGraph)
4745, 46syl 17 . . . . . 6 (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph β†’ βŸ¨π‘£, π‘’βŸ© ∈ UHGraph)
4828, 29cusgrsizeindb0 29319 . . . . . 6 ((βŸ¨π‘£, π‘’βŸ© ∈ UHGraph ∧ (β™―β€˜π‘£) = 0) β†’ (β™―β€˜(Edgβ€˜βŸ¨π‘£, π‘’βŸ©)) = ((β™―β€˜π‘£)C2))
4947, 48sylan 578 . . . . 5 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = 0) β†’ (β™―β€˜(Edgβ€˜βŸ¨π‘£, π‘’βŸ©)) = ((β™―β€˜π‘£)C2))
5044, 49eqtrd 2765 . . . 4 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = 0) β†’ (β™―β€˜ran 𝑒) = ((β™―β€˜π‘£)C2))
51 rnresi 6078 . . . . . . . . . 10 ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}) = {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}
5251fveq2i 6897 . . . . . . . . 9 (β™―β€˜ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) = (β™―β€˜{𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})
5341a1i 11 . . . . . . . . . . 11 (((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) β†’ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) = ran 𝑒)
5453rabeqdv 3435 . . . . . . . . . 10 (((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) β†’ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐})
5554fveq2d 6898 . . . . . . . . 9 (((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) β†’ (β™―β€˜{𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}) = (β™―β€˜{𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐}))
5652, 55eqtrid 2777 . . . . . . . 8 (((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) β†’ (β™―β€˜ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) = (β™―β€˜{𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐}))
5756eqeq1d 2727 . . . . . . 7 (((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) β†’ ((β™―β€˜ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2) ↔ (β™―β€˜{𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐}) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2)))
5857biimpd 228 . . . . . 6 (((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) β†’ ((β™―β€˜ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2) β†’ (β™―β€˜{𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐}) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2)))
5958imdistani 567 . . . . 5 ((((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (β™―β€˜ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2)) β†’ (((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (β™―β€˜{𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐}) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2)))
6041eqcomi 2734 . . . . . . 7 ran 𝑒 = (Edgβ€˜βŸ¨π‘£, π‘’βŸ©)
61 eqid 2725 . . . . . . 7 {𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐}
6228, 60, 61cusgrsize2inds 29323 . . . . . 6 ((𝑦 + 1) ∈ β„•0 β†’ ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) β†’ ((β™―β€˜{𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐}) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2) β†’ (β™―β€˜ran 𝑒) = ((β™―β€˜π‘£)C2))))
6362imp31 416 . . . . 5 ((((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (β™―β€˜{𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐}) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2)) β†’ (β™―β€˜ran 𝑒) = ((β™―β€˜π‘£)C2))
6459, 63syl 17 . . . 4 ((((𝑦 + 1) ∈ β„•0 ∧ (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (β™―β€˜ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐})) = ((β™―β€˜(𝑣 βˆ– {𝑛}))C2)) β†’ (β™―β€˜ran 𝑒) = ((β™―β€˜π‘£)C2))
6510, 14, 19, 24, 32, 39, 50, 64opfi1ind 14495 . . 3 ((βŸ¨π‘‰, (iEdgβ€˜πΊ)⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) β†’ (β™―β€˜ran (iEdgβ€˜πΊ)) = ((β™―β€˜π‘‰)C2))
669, 65sylan 578 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) β†’ (β™―β€˜ran (iEdgβ€˜πΊ)) = ((β™―β€˜π‘‰)C2))
675, 66eqtrd 2765 1 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) β†’ (β™―β€˜πΈ) = ((β™―β€˜π‘‰)C2))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βˆ‰ wnel 3036  {crab 3419  Vcvv 3463   βˆ– cdif 3942  {csn 4629  βŸ¨cop 4635   I cid 5574  ran crn 5678   β†Ύ cres 5679  β€˜cfv 6547  (class class class)co 7417  Fincfn 8962  0cc0 11138  1c1 11139   + caddc 11141  2c2 12297  β„•0cn0 12502  Ccbc 14293  β™―chash 14321  Vtxcvtx 28865  iEdgciedg 28866  Edgcedg 28916  UHGraphcuhgr 28925  USGraphcusgr 29018  ComplUSGraphccusgr 29279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7373  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-n0 12503  df-xnn0 12575  df-z 12589  df-uz 12853  df-rp 13007  df-fz 13517  df-seq 13999  df-fac 14265  df-bc 14294  df-hash 14322  df-vtx 28867  df-iedg 28868  df-edg 28917  df-uhgr 28927  df-upgr 28951  df-umgr 28952  df-uspgr 29019  df-usgr 29020  df-fusgr 29186  df-nbgr 29202  df-uvtx 29255  df-cplgr 29280  df-cusgr 29281
This theorem is referenced by:  fusgrmaxsize  29334
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