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

Theorem vdwmc 16850
Description: The predicate " The 𝑅, 𝑁-coloring 𝐹 contains a monochromatic AP of length 𝐾". (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdwmc.1 𝑋 ∈ V
vdwmc.2 (𝜑𝐾 ∈ ℕ0)
vdwmc.3 (𝜑𝐹:𝑋𝑅)
Assertion
Ref Expression
vdwmc (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
Distinct variable groups:   𝑎,𝑐,𝑑,𝐹   𝐾,𝑎,𝑐,𝑑   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑎,𝑑)   𝑅(𝑎,𝑐,𝑑)   𝑋(𝑎,𝑐,𝑑)

Proof of Theorem vdwmc
Dummy variables 𝑓 𝑘 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwmc.2 . . 3 (𝜑𝐾 ∈ ℕ0)
2 vdwmc.3 . . . 4 (𝜑𝐹:𝑋𝑅)
3 vdwmc.1 . . . 4 𝑋 ∈ V
4 fex 7176 . . . 4 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
52, 3, 4sylancl 586 . . 3 (𝜑𝐹 ∈ V)
6 fveq2 6842 . . . . . . . 8 (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
76rneqd 5893 . . . . . . 7 (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8 cnveq 5829 . . . . . . . . 9 (𝑓 = 𝐹𝑓 = 𝐹)
98imaeq1d 6012 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓 “ {𝑐}) = (𝐹 “ {𝑐}))
109pweqd 4577 . . . . . . 7 (𝑓 = 𝐹 → 𝒫 (𝑓 “ {𝑐}) = 𝒫 (𝐹 “ {𝑐}))
117, 10ineqan12d 4174 . . . . . 6 ((𝑘 = 𝐾𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
1211neeq1d 3003 . . . . 5 ((𝑘 = 𝐾𝑓 = 𝐹) → ((ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅ ↔ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
1312exbidv 1924 . . . 4 ((𝑘 = 𝐾𝑓 = 𝐹) → (∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅ ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
14 df-vdwmc 16841 . . . 4 MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅}
1513, 14brabga 5491 . . 3 ((𝐾 ∈ ℕ0𝐹 ∈ V) → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
161, 5, 15syl2anc 584 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
17 vdwapf 16844 . . . . 5 (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18 ffn 6668 . . . . 5 ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19 velpw 4565 . . . . . . 7 (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (𝐹 “ {𝑐}))
20 sseq1 3969 . . . . . . 7 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ⊆ (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2119, 20bitrid 282 . . . . . 6 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2221rexrn 7037 . . . . 5 ((AP‘𝐾) Fn (ℕ × ℕ) → (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
231, 17, 18, 224syl 19 . . . 4 (𝜑 → (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
24 elin 3926 . . . . . 6 (𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ↔ (𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
2524exbii 1850 . . . . 5 (∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
26 n0 4306 . . . . 5 ((ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
27 df-rex 3074 . . . . 5 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
2825, 26, 273bitr4ri 303 . . . 4 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅)
29 fveq2 6842 . . . . . . 7 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30 df-ov 7360 . . . . . . 7 (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
3129, 30eqtr4di 2794 . . . . . 6 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
3231sseq1d 3975 . . . . 5 (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3332rexxp 5798 . . . 4 (∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
3423, 28, 333bitr3g 312 . . 3 (𝜑 → ((ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3534exbidv 1924 . 2 (𝜑 → (∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3616, 35bitrd 278 1 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wne 2943  wrex 3073  Vcvv 3445  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586  cop 4592   class class class wbr 5105   × cxp 5631  ccnv 5632  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cn 12153  0cn0 12413  APcvdwa 16837   MonoAP cvdwm 16838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-vdwap 16840  df-vdwmc 16841
This theorem is referenced by:  vdwmc2  16851  vdwlem1  16853  vdwlem2  16854  vdwlem9  16861  vdwlem10  16862  vdwlem12  16864  vdwlem13  16865
  Copyright terms: Public domain W3C validator