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Theorem vdwmc 15961
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 6682 . . . 4 ((𝐹:𝑋𝑅𝑋 ∈ V) → 𝐹 ∈ V)
52, 3, 4sylancl 580 . . 3 (𝜑𝐹 ∈ V)
6 fveq2 6375 . . . . . . . 8 (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
76rneqd 5521 . . . . . . 7 (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8 cnveq 5464 . . . . . . . . 9 (𝑓 = 𝐹𝑓 = 𝐹)
98imaeq1d 5647 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓 “ {𝑐}) = (𝐹 “ {𝑐}))
109pweqd 4320 . . . . . . 7 (𝑓 = 𝐹 → 𝒫 (𝑓 “ {𝑐}) = 𝒫 (𝐹 “ {𝑐}))
117, 10ineqan12d 3978 . . . . . 6 ((𝑘 = 𝐾𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
1211neeq1d 2996 . . . . 5 ((𝑘 = 𝐾𝑓 = 𝐹) → ((ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅ ↔ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
1312exbidv 2016 . . . 4 ((𝑘 = 𝐾𝑓 = 𝐹) → (∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅ ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅))
14 df-vdwmc 15952 . . . 4 MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅}
1513, 14brabga 5150 . . 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 15955 . . . . 5 (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18 ffn 6223 . . . . 5 ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19 selpw 4322 . . . . . . 7 (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (𝐹 “ {𝑐}))
20 sseq1 3786 . . . . . . 7 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ⊆ (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2119, 20syl5bb 274 . . . . . 6 (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐})))
2221rexrn 6551 . . . . 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 3958 . . . . . 6 (𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ↔ (𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
2524exbii 1943 . . . . 5 (∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
26 n0 4095 . . . . 5 ((ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})))
27 df-rex 3061 . . . . 5 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (𝐹 “ {𝑐})))
2825, 26, 273bitr4ri 295 . . . 4 (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (𝐹 “ {𝑐}) ↔ (ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅)
29 fveq2 6375 . . . . . . 7 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30 df-ov 6845 . . . . . . 7 (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
3129, 30syl6eqr 2817 . . . . . 6 (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
3231sseq1d 3792 . . . . 5 (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3332rexxp 5433 . . . 4 (∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
3423, 28, 333bitr3g 304 . . 3 (𝜑 → ((ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3534exbidv 2016 . 2 (𝜑 → (∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
3616, 35bitrd 270 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 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  wrex 3056  Vcvv 3350  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334  cop 4340   class class class wbr 4809   × cxp 5275  ccnv 5276  ran crn 5278  cima 5280   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  cn 11274  0cn0 11538  APcvdwa 15948   MonoAP cvdwm 15949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-vdwap 15951  df-vdwmc 15952
This theorem is referenced by:  vdwmc2  15962  vdwlem1  15964  vdwlem2  15965  vdwlem9  15972  vdwlem10  15973  vdwlem12  15975  vdwlem13  15976
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