Theorem rexanuz 15384

Description: Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)

Assertion

Ref	Expression
rexanuz	⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))

Distinct variable groups:   𝑗,𝑘   𝜑,𝑗   𝜓,𝑗

Allowed substitution hints:   𝜑(𝑘)   𝜓(𝑘)

Proof of Theorem rexanuz

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.26 3111	. . . 4 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
2	1	rexbii 3094	. . 3 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
3		r19.40 3119	. . 3 ⊢ (∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
4	2, 3	sylbi 217	. 2 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
5		uzf 12881	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
6		ffn 6736	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
7		raleq 3323	. . . . 5 ⊢ (𝑥 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑥 𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
8	7	rexrn 7107	. . . 4 ⊢ (ℤ≥ Fn ℤ → (∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
9	5, 6, 8	mp2b 10	. . 3 ⊢ (∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)
10		raleq 3323	. . . . 5 ⊢ (𝑦 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑦 𝜓 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
11	10	rexrn 7107	. . . 4 ⊢ (ℤ≥ Fn ℤ → (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
12	5, 6, 11	mp2b 10	. . 3 ⊢ (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)
13		uzin2 15383	. . . . . . . . 9 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (𝑥 ∩ 𝑦) ∈ ran ℤ≥)
14		inss1 4237	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
15		ssralv 4052	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝑥 → (∀𝑘 ∈ 𝑥 𝜑 → ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜑))
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝑥 𝜑 → ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜑)
17		inss2 4238	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
18		ssralv 4052	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝑦 → (∀𝑘 ∈ 𝑦 𝜓 → ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜓))
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝑦 𝜓 → ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜓)
20	16, 19	anim12i 613	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝑥 𝜑 ∧ ∀𝑘 ∈ 𝑦 𝜓) → (∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜓))
21		r19.26 3111	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓) ↔ (∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜓))
22	20, 21	sylibr 234	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ 𝑥 𝜑 ∧ ∀𝑘 ∈ 𝑦 𝜓) → ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓))
23		raleq 3323	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓)))
24	23	rspcev 3622	. . . . . . . . 9 ⊢ (((𝑥 ∩ 𝑦) ∈ ran ℤ≥ ∧ ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓)) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
25	13, 22, 24	syl2an 596	. . . . . . . 8 ⊢ (((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) ∧ (∀𝑘 ∈ 𝑥 𝜑 ∧ ∀𝑘 ∈ 𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
26	25	an4s 660	. . . . . . 7 ⊢ (((𝑥 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑥 𝜑) ∧ (𝑦 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
27	26	rexlimdvaa 3156	. . . . . 6 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑥 𝜑) → (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓)))
28	27	rexlimiva 3147	. . . . 5 ⊢ (∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 → (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓)))
29	28	imp 406	. . . 4 ⊢ ((∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
30		raleq 3323	. . . . . 6 ⊢ (𝑧 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)))
31	30	rexrn 7107	. . . . 5 ⊢ (ℤ≥ Fn ℤ → (∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)))
32	5, 6, 31	mp2b 10	. . . 4 ⊢ (∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
33	29, 32	sylib 218	. . 3 ⊢ ((∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
34	9, 12, 33	syl2anbr 599	. 2 ⊢ ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
35	4, 34	impbii 209	1 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ran crn 5686   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  ℤcz 12613  ℤ_≥cuz 12878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-neg 11495  df-z 12614  df-uz 12879

This theorem is referenced by:  rexfiuz  15386  rexuz3  15387  rexanuz2  15388