Theorem vdwmc 16314

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	⊢ (𝜑 → 𝐾 ∈ ℕ0)
vdwmc.3	⊢ (𝜑 → 𝐹:𝑋⟶𝑅)

Assertion

Ref	Expression
vdwmc	⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑎 ∈ 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.

Step	Hyp	Ref	Expression
1		vdwmc.2	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
2		vdwmc.3	. . . 4 ⊢ (𝜑 → 𝐹:𝑋⟶𝑅)
3		vdwmc.1	. . . 4 ⊢ 𝑋 ∈ V
4		fex 6989	. . . 4 ⊢ ((𝐹:𝑋⟶𝑅 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
5	2, 3, 4	sylancl 588	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
6		fveq2 6670	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
7	6	rneqd 5808	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8		cnveq 5744	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
9	8	imaeq1d 5928	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {𝑐}) = (◡𝐹 “ {𝑐}))
10	9	pweqd 4558	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝒫 (◡𝑓 “ {𝑐}) = 𝒫 (◡𝐹 “ {𝑐}))
11	7, 10	ineqan12d 4191	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})))
12	11	neeq1d 3075	. . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
13	12	exbidv 1922	. . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
14		df-vdwmc 16305	. . . 4 ⊢ MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅}
15	13, 14	brabga 5421	. . 3 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐹 ∈ V) → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
16	1, 5, 15	syl2anc 586	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
17		vdwapf 16308	. . . . 5 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18		ffn 6514	. . . . 5 ⊢ ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19		velpw 4544	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (◡𝐹 “ {𝑐}))
20		sseq1 3992	. . . . . . 7 ⊢ (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ⊆ (◡𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
21	19, 20	syl5bb 285	. . . . . 6 ⊢ (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
22	21	rexrn 6853	. . . . 5 ⊢ ((AP‘𝐾) Fn (ℕ × ℕ) → (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ ∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
23	1, 17, 18, 22	4syl 19	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ ∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
24		elin 4169	. . . . . 6 ⊢ (𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ (𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
25	24	exbii 1848	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
26		n0 4310	. . . . 5 ⊢ ((ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})))
27		df-rex 3144	. . . . 5 ⊢ (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
28	25, 26, 27	3bitr4ri 306	. . . 4 ⊢ (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅)
29		fveq2 6670	. . . . . . 7 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30		df-ov 7159	. . . . . . 7 ⊢ (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
31	29, 30	syl6eqr 2874	. . . . . 6 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
32	31	sseq1d 3998	. . . . 5 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
33	32	rexxp 5713	. . . 4 ⊢ (∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
34	23, 28, 33	3bitr3g 315	. . 3 ⊢ (𝜑 → ((ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
35	34	exbidv 1922	. 2 ⊢ (𝜑 → (∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
36	16, 35	bitrd 281	1 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  ∃wrex 3139  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4567  ⟨cop 4573   class class class wbr 5066   × cxp 5553  ^◡ccnv 5554  ran crn 5556   “ cima 5558   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  ℕcn 11638  ℕ₀cn0 11898  APcvdwa 16301   MonoAP cvdwm 16302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-vdwap 16304  df-vdwmc 16305

This theorem is referenced by:  vdwmc2  16315  vdwlem1  16317  vdwlem2  16318  vdwlem9  16325  vdwlem10  16326  vdwlem12  16328  vdwlem13  16329