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Theorem sizeusglecusg 26014
 Description: The size of an undirected simple graph with 𝑛 vertices is at most the size of a complete simple graph with 𝑛 vertices (𝑛 may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Proof shortened by AV, 4-May-2021.)
Assertion
Ref Expression
sizeusglecusg ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹))

Proof of Theorem sizeusglecusg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 usgrav 25867 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
21simprd 478 . . . . . . . 8 (𝑉 USGrph 𝐸𝐸 ∈ V)
3 rnexg 6990 . . . . . . . 8 (𝐸 ∈ V → ran 𝐸 ∈ V)
4 resiexg 6994 . . . . . . . 8 (ran 𝐸 ∈ V → ( I ↾ ran 𝐸) ∈ V)
52, 3, 43syl 18 . . . . . . 7 (𝑉 USGrph 𝐸 → ( I ↾ ran 𝐸) ∈ V)
65adantr 480 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ( I ↾ ran 𝐸) ∈ V)
7 sizeusglecusglem1 26012 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐹)
8 f1eq1 6009 . . . . . 6 (𝑓 = ( I ↾ ran 𝐸) → (𝑓:ran 𝐸1-1→ran 𝐹 ↔ ( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐹))
96, 7, 8elabd 3321 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ∃𝑓 𝑓:ran 𝐸1-1→ran 𝐹)
109adantl 481 . . . 4 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹)) → ∃𝑓 𝑓:ran 𝐸1-1→ran 𝐹)
11 dmexg 6989 . . . . . . . 8 (𝐸 ∈ Fin → dom 𝐸 ∈ V)
1211adantr 480 . . . . . . 7 ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → dom 𝐸 ∈ V)
13 usgraf1 25889 . . . . . . . 8 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
1413adantr 480 . . . . . . 7 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → 𝐸:dom 𝐸1-1→ran 𝐸)
15 hashf1rn 13004 . . . . . . 7 ((dom 𝐸 ∈ V ∧ 𝐸:dom 𝐸1-1→ran 𝐸) → (#‘𝐸) = (#‘ran 𝐸))
1612, 14, 15syl2an 493 . . . . . 6 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹)) → (#‘𝐸) = (#‘ran 𝐸))
17 dmexg 6989 . . . . . . . 8 (𝐹 ∈ Fin → dom 𝐹 ∈ V)
1817adantl 481 . . . . . . 7 ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → dom 𝐹 ∈ V)
19 cusisusgra 25987 . . . . . . . . 9 (𝑉 ComplUSGrph 𝐹𝑉 USGrph 𝐹)
20 usgraf1 25889 . . . . . . . . 9 (𝑉 USGrph 𝐹𝐹:dom 𝐹1-1→ran 𝐹)
2119, 20syl 17 . . . . . . . 8 (𝑉 ComplUSGrph 𝐹𝐹:dom 𝐹1-1→ran 𝐹)
2221adantl 481 . . . . . . 7 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → 𝐹:dom 𝐹1-1→ran 𝐹)
23 hashf1rn 13004 . . . . . . 7 ((dom 𝐹 ∈ V ∧ 𝐹:dom 𝐹1-1→ran 𝐹) → (#‘𝐹) = (#‘ran 𝐹))
2418, 22, 23syl2an 493 . . . . . 6 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹)) → (#‘𝐹) = (#‘ran 𝐹))
2516, 24breq12d 4596 . . . . 5 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹)) → ((#‘𝐸) ≤ (#‘𝐹) ↔ (#‘ran 𝐸) ≤ (#‘ran 𝐹)))
26 rnfi 8132 . . . . . . . 8 (𝐸 ∈ Fin → ran 𝐸 ∈ Fin)
27 rnexg 6990 . . . . . . . 8 (𝐹 ∈ Fin → ran 𝐹 ∈ V)
2826, 27anim12i 588 . . . . . . 7 ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → (ran 𝐸 ∈ Fin ∧ ran 𝐹 ∈ V))
2928adantr 480 . . . . . 6 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹)) → (ran 𝐸 ∈ Fin ∧ ran 𝐹 ∈ V))
30 hashdom 13029 . . . . . 6 ((ran 𝐸 ∈ Fin ∧ ran 𝐹 ∈ V) → ((#‘ran 𝐸) ≤ (#‘ran 𝐹) ↔ ran 𝐸 ≼ ran 𝐹))
3129, 30syl 17 . . . . 5 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹)) → ((#‘ran 𝐸) ≤ (#‘ran 𝐹) ↔ ran 𝐸 ≼ ran 𝐹))
3227adantl 481 . . . . . . 7 ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → ran 𝐹 ∈ V)
33 brdomg 7851 . . . . . . 7 (ran 𝐹 ∈ V → (ran 𝐸 ≼ ran 𝐹 ↔ ∃𝑓 𝑓:ran 𝐸1-1→ran 𝐹))
3432, 33syl 17 . . . . . 6 ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → (ran 𝐸 ≼ ran 𝐹 ↔ ∃𝑓 𝑓:ran 𝐸1-1→ran 𝐹))
3534adantr 480 . . . . 5 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹)) → (ran 𝐸 ≼ ran 𝐹 ↔ ∃𝑓 𝑓:ran 𝐸1-1→ran 𝐹))
3625, 31, 353bitrd 293 . . . 4 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹)) → ((#‘𝐸) ≤ (#‘𝐹) ↔ ∃𝑓 𝑓:ran 𝐸1-1→ran 𝐹))
3710, 36mpbird 246 . . 3 (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹)) → (#‘𝐸) ≤ (#‘𝐹))
3837ex 449 . 2 ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹)))
39 usgrav 25867 . . . . . 6 (𝑉 USGrph 𝐹 → (𝑉 ∈ V ∧ 𝐹 ∈ V))
40 hashinf 12984 . . . . . . . . 9 ((𝐹 ∈ V ∧ ¬ 𝐹 ∈ Fin) → (#‘𝐹) = +∞)
41 hashxrcl 13010 . . . . . . . . . . 11 (𝐸 ∈ Fin → (#‘𝐸) ∈ ℝ*)
42 pnfge 11840 . . . . . . . . . . 11 ((#‘𝐸) ∈ ℝ* → (#‘𝐸) ≤ +∞)
4341, 42syl 17 . . . . . . . . . 10 (𝐸 ∈ Fin → (#‘𝐸) ≤ +∞)
44 breq2 4587 . . . . . . . . . 10 ((#‘𝐹) = +∞ → ((#‘𝐸) ≤ (#‘𝐹) ↔ (#‘𝐸) ≤ +∞))
4543, 44syl5ibr 235 . . . . . . . . 9 ((#‘𝐹) = +∞ → (𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹)))
4640, 45syl 17 . . . . . . . 8 ((𝐹 ∈ V ∧ ¬ 𝐹 ∈ Fin) → (𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹)))
4746ex 449 . . . . . . 7 (𝐹 ∈ V → (¬ 𝐹 ∈ Fin → (𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹))))
4847adantl 481 . . . . . 6 ((𝑉 ∈ V ∧ 𝐹 ∈ V) → (¬ 𝐹 ∈ Fin → (𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹))))
4919, 39, 483syl 18 . . . . 5 (𝑉 ComplUSGrph 𝐹 → (¬ 𝐹 ∈ Fin → (𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹))))
5049adantl 481 . . . 4 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (¬ 𝐹 ∈ Fin → (𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹))))
5150com13 86 . . 3 (𝐸 ∈ Fin → (¬ 𝐹 ∈ Fin → ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹))))
5251imp 444 . 2 ((𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹)))
53 sizeusglecusglem2 26013 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹𝐹 ∈ Fin) → 𝐸 ∈ Fin)
5453pm2.24d 146 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹𝐹 ∈ Fin) → (¬ 𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹)))
55543expia 1259 . . . 4 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (𝐹 ∈ Fin → (¬ 𝐸 ∈ Fin → (#‘𝐸) ≤ (#‘𝐹))))
5655com13 86 . . 3 𝐸 ∈ Fin → (𝐹 ∈ Fin → ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹))))
5756imp 444 . 2 ((¬ 𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹)))
58 hashinf 12984 . . . . . . . . . . . . . 14 ((𝐸 ∈ V ∧ ¬ 𝐸 ∈ Fin) → (#‘𝐸) = +∞)
59 pnfxr 9971 . . . . . . . . . . . . . . . . . . . 20 +∞ ∈ ℝ*
60 xrleid 11859 . . . . . . . . . . . . . . . . . . . 20 (+∞ ∈ ℝ* → +∞ ≤ +∞)
6159, 60mp1i 13 . . . . . . . . . . . . . . . . . . 19 (((#‘𝐸) = +∞ ∧ (#‘𝐹) = +∞) → +∞ ≤ +∞)
62 breq12 4588 . . . . . . . . . . . . . . . . . . 19 (((#‘𝐸) = +∞ ∧ (#‘𝐹) = +∞) → ((#‘𝐸) ≤ (#‘𝐹) ↔ +∞ ≤ +∞))
6361, 62mpbird 246 . . . . . . . . . . . . . . . . . 18 (((#‘𝐸) = +∞ ∧ (#‘𝐹) = +∞) → (#‘𝐸) ≤ (#‘𝐹))
6463expcom 450 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) = +∞ → ((#‘𝐸) = +∞ → (#‘𝐸) ≤ (#‘𝐹)))
6540, 64syl 17 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ V ∧ ¬ 𝐹 ∈ Fin) → ((#‘𝐸) = +∞ → (#‘𝐸) ≤ (#‘𝐹)))
6665ex 449 . . . . . . . . . . . . . . 15 (𝐹 ∈ V → (¬ 𝐹 ∈ Fin → ((#‘𝐸) = +∞ → (#‘𝐸) ≤ (#‘𝐹))))
6766com13 86 . . . . . . . . . . . . . 14 ((#‘𝐸) = +∞ → (¬ 𝐹 ∈ Fin → (𝐹 ∈ V → (#‘𝐸) ≤ (#‘𝐹))))
6858, 67syl 17 . . . . . . . . . . . . 13 ((𝐸 ∈ V ∧ ¬ 𝐸 ∈ Fin) → (¬ 𝐹 ∈ Fin → (𝐹 ∈ V → (#‘𝐸) ≤ (#‘𝐹))))
6968expcom 450 . . . . . . . . . . . 12 𝐸 ∈ Fin → (𝐸 ∈ V → (¬ 𝐹 ∈ Fin → (𝐹 ∈ V → (#‘𝐸) ≤ (#‘𝐹)))))
7069com23 84 . . . . . . . . . . 11 𝐸 ∈ Fin → (¬ 𝐹 ∈ Fin → (𝐸 ∈ V → (𝐹 ∈ V → (#‘𝐸) ≤ (#‘𝐹)))))
7170imp 444 . . . . . . . . . 10 ((¬ 𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → (𝐸 ∈ V → (𝐹 ∈ V → (#‘𝐸) ≤ (#‘𝐹))))
7271com13 86 . . . . . . . . 9 (𝐹 ∈ V → (𝐸 ∈ V → ((¬ 𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → (#‘𝐸) ≤ (#‘𝐹))))
7372adantl 481 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐹 ∈ V) → (𝐸 ∈ V → ((¬ 𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → (#‘𝐸) ≤ (#‘𝐹))))
7419, 39, 733syl 18 . . . . . . 7 (𝑉 ComplUSGrph 𝐹 → (𝐸 ∈ V → ((¬ 𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → (#‘𝐸) ≤ (#‘𝐹))))
7574com12 32 . . . . . 6 (𝐸 ∈ V → (𝑉 ComplUSGrph 𝐹 → ((¬ 𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → (#‘𝐸) ≤ (#‘𝐹))))
7675adantl 481 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ComplUSGrph 𝐹 → ((¬ 𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → (#‘𝐸) ≤ (#‘𝐹))))
771, 76syl 17 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 ComplUSGrph 𝐹 → ((¬ 𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → (#‘𝐸) ≤ (#‘𝐹))))
7877imp 444 . . 3 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ((¬ 𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → (#‘𝐸) ≤ (#‘𝐹)))
7978com12 32 . 2 ((¬ 𝐸 ∈ Fin ∧ ¬ 𝐹 ∈ Fin) → ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹)))
8038, 52, 57, 794cases 987 1 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040  –1-1→wf1 5801  ‘cfv 5804   ≼ cdom 7839  Fincfn 7841  +∞cpnf 9950  ℝ*cxr 9952   ≤ cle 9954  #chash 12979   USGrph cusg 25859   ComplUSGrph ccusgra 25947 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-cusgra 25950 This theorem is referenced by:  usgramaxsize  26015
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