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Theorem vdwmc 16781
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 7167 . . . 4 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
52, 3, 4sylancl 587 . . 3 (𝜑𝐹 ∈ V)
6 fveq2 6834 . . . . . . . 8 (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
76rneqd 5886 . . . . . . 7 (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8 cnveq 5822 . . . . . . . . 9 (𝑓 = 𝐹𝑓 = 𝐹)
98imaeq1d 6005 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓 “ {𝑐}) = (𝐹 “ {𝑐}))
109pweqd 4572 . . . . . . 7 (𝑓 = 𝐹 → 𝒫 (𝑓 “ {𝑐}) = 𝒫 (𝐹 “ {𝑐}))
117, 10ineqan12d 4169 . . . . . 6 ((𝑘 = 𝐾𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
1211neeq1d 3001 . . . . 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 16772 . . . 4 MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅}
1513, 14brabga 5485 . . 3 ((𝐾 ∈ ℕ0𝐹 ∈ V) → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
161, 5, 15syl2anc 585 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
17 vdwapf 16775 . . . . 5 (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18 ffn 6660 . . . . 5 ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19 velpw 4560 . . . . . . 7 (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (𝐹 “ {𝑐}))
20 sseq1 3964 . . . . . . 7 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ⊆ (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2119, 20bitrid 283 . . . . . 6 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2221rexrn 7028 . . . . 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 3921 . . . . . 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 4301 . . . . 5 ((ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
27 df-rex 3072 . . . . 5 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
2825, 26, 273bitr4ri 304 . . . 4 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅)
29 fveq2 6834 . . . . . . 7 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30 df-ov 7349 . . . . . . 7 (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
3129, 30eqtr4di 2795 . . . . . 6 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
3231sseq1d 3970 . . . . 5 (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3332rexxp 5791 . . . 4 (∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
3423, 28, 333bitr3g 313 . . 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 279 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 397   = wceq 1541  wex 1781  wcel 2106  wne 2941  wrex 3071  Vcvv 3443  cin 3904  wss 3905  c0 4277  𝒫 cpw 4555  {csn 4581  cop 4587   class class class wbr 5100   × cxp 5625  ccnv 5626  ran crn 5628  cima 5630   Fn wfn 6483  wf 6484  cfv 6488  (class class class)co 7346  cn 12083  0cn0 12343  APcvdwa 16768   MonoAP cvdwm 16769
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 2708  ax-rep 5237  ax-sep 5251  ax-nul 5258  ax-pow 5315  ax-pr 5379  ax-un 7659  ax-cnex 11037  ax-resscn 11038  ax-1cn 11039  ax-icn 11040  ax-addcl 11041  ax-addrcl 11042  ax-mulcl 11043  ax-mulrcl 11044  ax-mulcom 11045  ax-addass 11046  ax-mulass 11047  ax-distr 11048  ax-i2m1 11049  ax-1ne0 11050  ax-1rid 11051  ax-rnegex 11052  ax-rrecex 11053  ax-cnre 11054  ax-pre-lttri 11055  ax-pre-lttrn 11056  ax-pre-ltadd 11057  ax-pre-mulgt0 11058
This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3924  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5184  df-tr 5218  df-id 5525  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5582  df-we 5584  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-pred 6246  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7302  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7790  df-1st 7908  df-2nd 7909  df-frecs 8176  df-wrecs 8207  df-recs 8281  df-rdg 8320  df-er 8578  df-en 8814  df-dom 8815  df-sdom 8816  df-pnf 11121  df-mnf 11122  df-xr 11123  df-ltxr 11124  df-le 11125  df-sub 11317  df-neg 11318  df-nn 12084  df-n0 12344  df-z 12430  df-uz 12693  df-fz 13350  df-vdwap 16771  df-vdwmc 16772
This theorem is referenced by:  vdwmc2  16782  vdwlem1  16784  vdwlem2  16785  vdwlem9  16792  vdwlem10  16793  vdwlem12  16795  vdwlem13  16796
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