Theorem vdwmc 17038

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 7225	. . . 4 ⊢ ((𝐹:𝑋⟶𝑅 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
5	2, 3, 4	sylancl 597	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
6		fveq2 6882	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
7	6	rneqd 5929	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8		cnveq 5860	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
9	8	imaeq1d 6062	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {𝑐}) = (◡𝐹 “ {𝑐}))
10	9	pweqd 4584	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝒫 (◡𝑓 “ {𝑐}) = 𝒫 (◡𝐹 “ {𝑐}))
11	7, 10	ineqan12d 4183	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})))
12	11	neeq1d 3023	. . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
13	12	exbidv 1948	. . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
14		df-vdwmc 17029	. . . 4 ⊢ MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅}
15	13, 14	brabga 5519	. . 3 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐹 ∈ V) → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
16	1, 5, 15	syl2anc 595	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
17		vdwapf 17032	. . . . 5 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18		ffn 6706	. . . . 5 ⊢ ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19		velpw 4572	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (◡𝐹 “ {𝑐}))
20		sseq1 3970	. . . . . . 7 ⊢ (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ⊆ (◡𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
21	19, 20	bitrid 286	. . . . . 6 ⊢ (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
22	21	rexrn 7083	. . . . 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 20	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ ∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
24		elin 3929	. . . . . 6 ⊢ (𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ (𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
25	24	exbii 1875	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
26		n0 4315	. . . . 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})))
28	25, 26, 27	3bitr4ri 307	. . . 4 ⊢ (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅)
29		fveq2 6882	. . . . . . 7 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30		df-ov 7414	. . . . . . 7 ⊢ (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
31	29, 30	eqtr4di 2822	. . . . . 6 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
32	31	sseq1d 3976	. . . . 5 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
33	32	rexxp 5829	. . . 4 ⊢ (∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
34	23, 28, 33	3bitr3g 316	. . 3 ⊢ (𝜑 → ((ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
35	34	exbidv 1948	. 2 ⊢ (𝜑 → (∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
36	16, 35	bitrd 282	1 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  ⟨cop 4600   class class class wbr 5113   × cxp 5660  ^◡ccnv 5661  ran crn 5663   “ cima 5665   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ℕcn 12233  ℕ₀cn0 12504  APcvdwa 17025   MonoAP cvdwm 17026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-vdwap 17028  df-vdwmc 17029

This theorem is referenced by:  vdwmc2  17039  vdwlem1  17041  vdwlem2  17042  vdwlem9  17049  vdwlem10  17050  vdwlem12  17052  vdwlem13  17053