MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgrsize Structured version   Visualization version   GIF version

Theorem cusgrsize 29255
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 28849 . . . . 5 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
31, 2eqtri 2755 . . . 4 𝐸 = ran (iEdgβ€˜πΊ)
43a1i 11 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) β†’ 𝐸 = ran (iEdgβ€˜πΊ))
54fveq2d 6895 . 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 4872 . . . 4 βŸ¨π‘‰, (iEdgβ€˜πΊ)⟩ = ⟨(Vtxβ€˜πΊ), (iEdgβ€˜πΊ)⟩
8 cusgrop 29238 . . . 4 (𝐺 ∈ ComplUSGraph β†’ ⟨(Vtxβ€˜πΊ), (iEdgβ€˜πΊ)⟩ ∈ ComplUSGraph)
97, 8eqeltrid 2832 . . 3 (𝐺 ∈ ComplUSGraph β†’ βŸ¨π‘‰, (iEdgβ€˜πΊ)⟩ ∈ ComplUSGraph)
10 fvex 6904 . . . 4 (iEdgβ€˜πΊ) ∈ V
11 fvex 6904 . . . . 5 (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∈ V
12 rabexg 5327 . . . . . 6 ((Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∈ V β†’ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐} ∈ V)
1312resiexd 7222 . . . . 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 5932 . . . . . 6 (𝑒 = (iEdgβ€˜πΊ) β†’ ran 𝑒 = ran (iEdgβ€˜πΊ))
1615fveq2d 6895 . . . . 5 (𝑒 = (iEdgβ€˜πΊ) β†’ (β™―β€˜ran 𝑒) = (β™―β€˜ran (iEdgβ€˜πΊ)))
17 fveq2 6891 . . . . . 6 (𝑣 = 𝑉 β†’ (β™―β€˜π‘£) = (β™―β€˜π‘‰))
1817oveq1d 7429 . . . . 5 (𝑣 = 𝑉 β†’ ((β™―β€˜π‘£)C2) = ((β™―β€˜π‘‰)C2))
1916, 18eqeqan12rd 2742 . . . 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 5932 . . . . . 6 (𝑒 = 𝑓 β†’ ran 𝑒 = ran 𝑓)
2120fveq2d 6895 . . . . 5 (𝑒 = 𝑓 β†’ (β™―β€˜ran 𝑒) = (β™―β€˜ran 𝑓))
22 fveq2 6891 . . . . . 6 (𝑣 = 𝑀 β†’ (β™―β€˜π‘£) = (β™―β€˜π‘€))
2322oveq1d 7429 . . . . 5 (𝑣 = 𝑀 β†’ ((β™―β€˜π‘£)C2) = ((β™―β€˜π‘€)C2))
2421, 23eqeqan12rd 2742 . . . 4 ((𝑣 = 𝑀 ∧ 𝑒 = 𝑓) β†’ ((β™―β€˜ran 𝑒) = ((β™―β€˜π‘£)C2) ↔ (β™―β€˜ran 𝑓) = ((β™―β€˜π‘€)C2)))
25 vex 3473 . . . . . . 7 𝑣 ∈ V
26 vex 3473 . . . . . . 7 𝑒 ∈ V
2725, 26opvtxfvi 28809 . . . . . 6 (Vtxβ€˜βŸ¨π‘£, π‘’βŸ©) = 𝑣
2827eqcomi 2736 . . . . 5 𝑣 = (Vtxβ€˜βŸ¨π‘£, π‘’βŸ©)
29 eqid 2727 . . . . 5 (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) = (Edgβ€˜βŸ¨π‘£, π‘’βŸ©)
30 eqid 2727 . . . . 5 {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐} = {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}
31 eqid 2727 . . . . 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 29249 . . . 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 5932 . . . . . . 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 6895 . . . . . 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 481 . . . . 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 6891 . . . . . . 7 (𝑀 = (𝑣 βˆ– {𝑛}) β†’ (β™―β€˜π‘€) = (β™―β€˜(𝑣 βˆ– {𝑛})))
3736adantr 480 . . . . . 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 7429 . . . . 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 2743 . . . 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 28851 . . . . . . . . 9 ((𝑣 ∈ V ∧ 𝑒 ∈ V) β†’ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) = ran 𝑒)
4140el2v 3477 . . . . . . . 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 2733 . . . . . 6 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = 0) β†’ ran 𝑒 = (Edgβ€˜βŸ¨π‘£, π‘’βŸ©))
4443fveq2d 6895 . . . . 5 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = 0) β†’ (β™―β€˜ran 𝑒) = (β™―β€˜(Edgβ€˜βŸ¨π‘£, π‘’βŸ©)))
45 cusgrusgr 29219 . . . . . . 7 (βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph β†’ βŸ¨π‘£, π‘’βŸ© ∈ USGraph)
46 usgruhgr 28986 . . . . . . 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 29250 . . . . . 6 ((βŸ¨π‘£, π‘’βŸ© ∈ UHGraph ∧ (β™―β€˜π‘£) = 0) β†’ (β™―β€˜(Edgβ€˜βŸ¨π‘£, π‘’βŸ©)) = ((β™―β€˜π‘£)C2))
4947, 48sylan 579 . . . . 5 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = 0) β†’ (β™―β€˜(Edgβ€˜βŸ¨π‘£, π‘’βŸ©)) = ((β™―β€˜π‘£)C2))
5044, 49eqtrd 2767 . . . 4 ((βŸ¨π‘£, π‘’βŸ© ∈ ComplUSGraph ∧ (β™―β€˜π‘£) = 0) β†’ (β™―β€˜ran 𝑒) = ((β™―β€˜π‘£)C2))
51 rnresi 6072 . . . . . . . . . 10 ran ( I β†Ύ {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}) = {𝑐 ∈ (Edgβ€˜βŸ¨π‘£, π‘’βŸ©) ∣ 𝑛 βˆ‰ 𝑐}
5251fveq2i 6894 . . . . . . . . 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 3442 . . . . . . . . . 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 6895 . . . . . . . . 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 2779 . . . . . . . 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 2729 . . . . . . 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 568 . . . . 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 2736 . . . . . . 7 ran 𝑒 = (Edgβ€˜βŸ¨π‘£, π‘’βŸ©)
61 eqid 2727 . . . . . . 7 {𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 βˆ‰ 𝑐}
6228, 60, 61cusgrsize2inds 29254 . . . . . 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 417 . . . . 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 14487 . . 3 ((βŸ¨π‘‰, (iEdgβ€˜πΊ)⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) β†’ (β™―β€˜ran (iEdgβ€˜πΊ)) = ((β™―β€˜π‘‰)C2))
669, 65sylan 579 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) β†’ (β™―β€˜ran (iEdgβ€˜πΊ)) = ((β™―β€˜π‘‰)C2))
675, 66eqtrd 2767 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   βˆ‰ wnel 3041  {crab 3427  Vcvv 3469   βˆ– cdif 3941  {csn 4624  βŸ¨cop 4630   I cid 5569  ran crn 5673   β†Ύ cres 5674  β€˜cfv 6542  (class class class)co 7414  Fincfn 8955  0cc0 11130  1c1 11131   + caddc 11133  2c2 12289  β„•0cn0 12494  Ccbc 14285  β™―chash 14313  Vtxcvtx 28796  iEdgciedg 28797  Edgcedg 28847  UHGraphcuhgr 28856  USGraphcusgr 28949  ComplUSGraphccusgr 29210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-fz 13509  df-seq 13991  df-fac 14257  df-bc 14286  df-hash 14314  df-vtx 28798  df-iedg 28799  df-edg 28848  df-uhgr 28858  df-upgr 28882  df-umgr 28883  df-uspgr 28950  df-usgr 28951  df-fusgr 29117  df-nbgr 29133  df-uvtx 29186  df-cplgr 29211  df-cusgr 29212
This theorem is referenced by:  fusgrmaxsize  29265
  Copyright terms: Public domain W3C validator