Theorem vdwmc 16908

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 587	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
6		fveq2 6889	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
7	6	rneqd 5936	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8		cnveq 5872	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
9	8	imaeq1d 6057	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {𝑐}) = (◡𝐹 “ {𝑐}))
10	9	pweqd 4619	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝒫 (◡𝑓 “ {𝑐}) = 𝒫 (◡𝐹 “ {𝑐}))
11	7, 10	ineqan12d 4214	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})))
12	11	neeq1d 3001	. . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
13	12	exbidv 1925	. . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
14		df-vdwmc 16899	. . . 4 ⊢ MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅}
15	13, 14	brabga 5534	. . 3 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐹 ∈ V) → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
16	1, 5, 15	syl2anc 585	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
17		vdwapf 16902	. . . . 5 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18		ffn 6715	. . . . 5 ⊢ ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19		velpw 4607	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (◡𝐹 “ {𝑐}))
20		sseq1 4007	. . . . . . 7 ⊢ (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ⊆ (◡𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
21	19, 20	bitrid 283	. . . . . 6 ⊢ (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
22	21	rexrn 7086	. . . . 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 3964	. . . . . 6 ⊢ (𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ (𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
25	24	exbii 1851	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
26		n0 4346	. . . . 5 ⊢ ((ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})))
27		df-rex 3072	. . . . 5 ⊢ (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
28	25, 26, 27	3bitr4ri 304	. . . 4 ⊢ (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅)
29		fveq2 6889	. . . . . . 7 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30		df-ov 7409	. . . . . . 7 ⊢ (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
31	29, 30	eqtr4di 2791	. . . . . 6 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
32	31	sseq1d 4013	. . . . 5 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
33	32	rexxp 5841	. . . 4 ⊢ (∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
34	23, 28, 33	3bitr3g 313	. . 3 ⊢ (𝜑 → ((ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
35	34	exbidv 1925	. 2 ⊢ (𝜑 → (∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
36	16, 35	bitrd 279	1 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  ∃wrex 3071  Vcvv 3475   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  {csn 4628  ⟨cop 4634   class class class wbr 5148   × cxp 5674  ^◡ccnv 5675  ran crn 5677   “ cima 5679   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406  ℕcn 12209  ℕ₀cn0 12469  APcvdwa 16895   MonoAP cvdwm 16896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-vdwap 16898  df-vdwmc 16899

This theorem is referenced by:  vdwmc2  16909  vdwlem1  16911  vdwlem2  16912  vdwlem9  16919  vdwlem10  16920  vdwlem12  16922  vdwlem13  16923