Theorem vdwmc 17033

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 7224	. . . 4 ⊢ ((𝐹:𝑋⟶𝑅 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
5	2, 3, 4	sylancl 597	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
6		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
7	6	rneqd 5928	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8		cnveq 5859	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
9	8	imaeq1d 6061	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {𝑐}) = (◡𝐹 “ {𝑐}))
10	9	pweqd 4579	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝒫 (◡𝑓 “ {𝑐}) = 𝒫 (◡𝐹 “ {𝑐}))
11	7, 10	ineqan12d 4175	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})))
12	11	neeq1d 3017	. . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
13	12	exbidv 1951	. . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
14		df-vdwmc 17024	. . . 4 ⊢ MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅}
15	13, 14	brabga 5518	. . 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 17027	. . . . 5 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18		ffn 6705	. . . . 5 ⊢ ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19		velpw 4567	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (◡𝐹 “ {𝑐}))
20		sseq1 3962	. . . . . . 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 7082	. . . . 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 3921	. . . . . 6 ⊢ (𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ (𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
25	24	exbii 1878	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
26		n0 4307	. . . . 5 ⊢ ((ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})))
27		df-rex 3090	. . . . 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 6881	. . . . . . 7 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30		df-ov 7413	. . . . . . 7 ⊢ (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
31	29, 30	eqtr4di 2816	. . . . . 6 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
32	31	sseq1d 3968	. . . . 5 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
33	32	rexxp 5828	. . . 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 1951	. 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 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589  ⟨cop 4595   class class class wbr 5109   × cxp 5659  ^◡ccnv 5660  ran crn 5662   “ cima 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℕcn 12228  ℕ₀cn0 12499  APcvdwa 17020   MonoAP cvdwm 17021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-vdwap 17023  df-vdwmc 17024

This theorem is referenced by:  vdwmc2  17034  vdwlem1  17036  vdwlem2  17037  vdwlem9  17044  vdwlem10  17045  vdwlem12  17047  vdwlem13  17048