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Theorem vdwmc 15682
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 6490 . . . 4 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
52, 3, 4sylancl 694 . . 3 (𝜑𝐹 ∈ V)
6 fveq2 6191 . . . . . . . 8 (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
76rneqd 5353 . . . . . . 7 (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8 cnveq 5296 . . . . . . . . 9 (𝑓 = 𝐹𝑓 = 𝐹)
98imaeq1d 5465 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓 “ {𝑐}) = (𝐹 “ {𝑐}))
109pweqd 4163 . . . . . . 7 (𝑓 = 𝐹 → 𝒫 (𝑓 “ {𝑐}) = 𝒫 (𝐹 “ {𝑐}))
117, 10ineqan12d 3816 . . . . . 6 ((𝑘 = 𝐾𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
1211neeq1d 2853 . . . . 5 ((𝑘 = 𝐾𝑓 = 𝐹) → ((ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅ ↔ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
1312exbidv 1850 . . . 4 ((𝑘 = 𝐾𝑓 = 𝐹) → (∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅ ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
14 df-vdwmc 15673 . . . 4 MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅}
1513, 14brabga 4989 . . 3 ((𝐾 ∈ ℕ0𝐹 ∈ V) → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
161, 5, 15syl2anc 693 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
17 vdwapf 15676 . . . . 5 (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18 ffn 6045 . . . . 5 ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19 selpw 4165 . . . . . . 7 (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (𝐹 “ {𝑐}))
20 sseq1 3626 . . . . . . 7 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ⊆ (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2119, 20syl5bb 272 . . . . . 6 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2221rexrn 6361 . . . . 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 3796 . . . . . 6 (𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ↔ (𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
2524exbii 1774 . . . . 5 (∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
26 n0 3931 . . . . 5 ((ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
27 df-rex 2918 . . . . 5 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
2825, 26, 273bitr4ri 293 . . . 4 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅)
29 fveq2 6191 . . . . . . 7 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30 df-ov 6653 . . . . . . 7 (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
3129, 30syl6eqr 2674 . . . . . 6 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
3231sseq1d 3632 . . . . 5 (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3332rexxp 5264 . . . 4 (∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
3423, 28, 333bitr3g 302 . . 3 (𝜑 → ((ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3534exbidv 1850 . 2 (𝜑 → (∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3616, 35bitrd 268 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 196  wa 384   = wceq 1483  wex 1704  wcel 1990  wne 2794  wrex 2913  Vcvv 3200  cin 3573  wss 3574  c0 3915  𝒫 cpw 4158  {csn 4177  cop 4183   class class class wbr 4653   × cxp 5112  ccnv 5113  ran crn 5115  cima 5117   Fn wfn 5883  wf 5884  cfv 5888  (class class class)co 6650  cn 11020  0cn0 11292  APcvdwa 15669   MonoAP cvdwm 15670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-vdwap 15672  df-vdwmc 15673
This theorem is referenced by:  vdwmc2  15683  vdwlem1  15685  vdwlem2  15686  vdwlem9  15693  vdwlem10  15694  vdwlem12  15696  vdwlem13  15697
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