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Theorem vdwmc 16086
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 6761 . . . 4 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
52, 3, 4sylancl 580 . . 3 (𝜑𝐹 ∈ V)
6 fveq2 6446 . . . . . . . 8 (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
76rneqd 5598 . . . . . . 7 (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8 cnveq 5541 . . . . . . . . 9 (𝑓 = 𝐹𝑓 = 𝐹)
98imaeq1d 5719 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓 “ {𝑐}) = (𝐹 “ {𝑐}))
109pweqd 4384 . . . . . . 7 (𝑓 = 𝐹 → 𝒫 (𝑓 “ {𝑐}) = 𝒫 (𝐹 “ {𝑐}))
117, 10ineqan12d 4039 . . . . . 6 ((𝑘 = 𝐾𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
1211neeq1d 3028 . . . . 5 ((𝑘 = 𝐾𝑓 = 𝐹) → ((ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅ ↔ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
1312exbidv 1964 . . . 4 ((𝑘 = 𝐾𝑓 = 𝐹) → (∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅ ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
14 df-vdwmc 16077 . . . 4 MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅}
1513, 14brabga 5226 . . 3 ((𝐾 ∈ ℕ0𝐹 ∈ V) → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
161, 5, 15syl2anc 579 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
17 vdwapf 16080 . . . . 5 (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18 ffn 6291 . . . . 5 ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19 selpw 4386 . . . . . . 7 (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (𝐹 “ {𝑐}))
20 sseq1 3845 . . . . . . 7 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ⊆ (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2119, 20syl5bb 275 . . . . . 6 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2221rexrn 6625 . . . . 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 4019 . . . . . 6 (𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ↔ (𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
2524exbii 1892 . . . . 5 (∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
26 n0 4159 . . . . 5 ((ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
27 df-rex 3096 . . . . 5 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
2825, 26, 273bitr4ri 296 . . . 4 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅)
29 fveq2 6446 . . . . . . 7 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30 df-ov 6925 . . . . . . 7 (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
3129, 30syl6eqr 2832 . . . . . 6 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
3231sseq1d 3851 . . . . 5 (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3332rexxp 5510 . . . 4 (∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
3423, 28, 333bitr3g 305 . . 3 (𝜑 → ((ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3534exbidv 1964 . 2 (𝜑 → (∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3616, 35bitrd 271 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 198  wa 386   = wceq 1601  wex 1823  wcel 2107  wne 2969  wrex 3091  Vcvv 3398  cin 3791  wss 3792  c0 4141  𝒫 cpw 4379  {csn 4398  cop 4404   class class class wbr 4886   × cxp 5353  ccnv 5354  ran crn 5356  cima 5358   Fn wfn 6130  wf 6131  cfv 6135  (class class class)co 6922  cn 11374  0cn0 11642  APcvdwa 16073   MonoAP cvdwm 16074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-vdwap 16076  df-vdwmc 16077
This theorem is referenced by:  vdwmc2  16087  vdwlem1  16089  vdwlem2  16090  vdwlem9  16097  vdwlem10  16098  vdwlem12  16100  vdwlem13  16101
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