Theorem vdwmc 16086

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 6761	. . . 4 ⊢ ((𝐹:𝑋⟶𝑅 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
5	2, 3, 4	sylancl 580	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
6		fveq2 6446	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (AP‘𝑘) = (AP‘𝐾))
7	6	rneqd 5598	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ran (AP‘𝑘) = ran (AP‘𝐾))
8		cnveq 5541	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
9	8	imaeq1d 5719	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {𝑐}) = (◡𝐹 “ {𝑐}))
10	9	pweqd 4384	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝒫 (◡𝑓 “ {𝑐}) = 𝒫 (◡𝐹 “ {𝑐}))
11	7, 10	ineqan12d 4039	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) = (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})))
12	11	neeq1d 3028	. . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → ((ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
13	12	exbidv 1964	. . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑓 = 𝐹) → (∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅ ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
14		df-vdwmc 16077	. . . 4 ⊢ MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅}
15	13, 14	brabga 5226	. . 3 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐹 ∈ V) → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
16	1, 5, 15	syl2anc 579	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅))
17		vdwapf 16080	. . . . 5 ⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
18		ffn 6291	. . . . 5 ⊢ ((AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ → (AP‘𝐾) Fn (ℕ × ℕ))
19		selpw 4386	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ 𝑧 ⊆ (◡𝐹 “ {𝑐}))
20		sseq1 3845	. . . . . . 7 ⊢ (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ⊆ (◡𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
21	19, 20	syl5bb 275	. . . . . 6 ⊢ (𝑧 = ((AP‘𝐾)‘𝑤) → (𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ ((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐})))
22	21	rexrn 6625	. . . . 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 4019	. . . . . 6 ⊢ (𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ (𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
25	24	exbii 1892	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ↔ ∃𝑧(𝑧 ∈ ran (AP‘𝐾) ∧ 𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐})))
26		n0 4159	. . . . 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 296	. . . 4 ⊢ (∃𝑧 ∈ ran (AP‘𝐾)𝑧 ∈ 𝒫 (◡𝐹 “ {𝑐}) ↔ (ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅)
29		fveq2 6446	. . . . . . 7 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩))
30		df-ov 6925	. . . . . . 7 ⊢ (𝑎(AP‘𝐾)𝑑) = ((AP‘𝐾)‘⟨𝑎, 𝑑⟩)
31	29, 30	syl6eqr 2832	. . . . . 6 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → ((AP‘𝐾)‘𝑤) = (𝑎(AP‘𝐾)𝑑))
32	31	sseq1d 3851	. . . . 5 ⊢ (𝑤 = ⟨𝑎, 𝑑⟩ → (((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
33	32	rexxp 5510	. . . 4 ⊢ (∃𝑤 ∈ (ℕ × ℕ)((AP‘𝐾)‘𝑤) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
34	23, 28, 33	3bitr3g 305	. . 3 ⊢ (𝜑 → ((ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
35	34	exbidv 1964	. 2 ⊢ (𝜑 → (∃𝑐(ran (AP‘𝐾) ∩ 𝒫 (◡𝐹 “ {𝑐})) ≠ ∅ ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
36	16, 35	bitrd 271	1 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2107   ≠ wne 2969  ∃wrex 3091  Vcvv 3398   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  𝒫 cpw 4379  {csn 4398  ⟨cop 4404   class class class wbr 4886   × cxp 5353  ^◡ccnv 5354  ran crn 5356   “ cima 5358   Fn wfn 6130  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922  ℕcn 11374  ℕ₀cn0 11642  APcvdwa 16073   MonoAP cvdwm 16074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-vdwap 16076  df-vdwmc 16077

This theorem is referenced by:  vdwmc2  16087  vdwlem1  16089  vdwlem2  16090  vdwlem9  16097  vdwlem10  16098  vdwlem12  16100  vdwlem13  16101