Theorem rexanuz 15437

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 3124	. . . 4 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
2	1	rexbii 3111	. . 3 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
3		r19.40 3130	. . 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 220	. 2 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
5		uzf 12894	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
6		ffn 6706	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
7		raleq 3318	. . . . 5 ⊢ (𝑥 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑥 𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
8	7	rexrn 7084	. . . 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 3318	. . . . 5 ⊢ (𝑦 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑦 𝜓 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
11	10	rexrn 7084	. . . 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 15436	. . . . . . . . 9 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (𝑥 ∩ 𝑦) ∈ ran ℤ≥)
14		inss1 4185	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
15		ssralv 4003	. . . . . . . . . . . 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 4186	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
18		ssralv 4003	. . . . . . . . . . . 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 625	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝑥 𝜑 ∧ ∀𝑘 ∈ 𝑦 𝜓) → (∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜓))
21		r19.26 3124	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓) ↔ (∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜓))
22	20, 21	sylibr 237	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ 𝑥 𝜑 ∧ ∀𝑘 ∈ 𝑦 𝜓) → ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓))
23		raleq 3318	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓)))
24	23	rspcev 3579	. . . . . . . . 9 ⊢ (((𝑥 ∩ 𝑦) ∈ ran ℤ≥ ∧ ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓)) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
25	13, 22, 24	syl2an 608	. . . . . . . 8 ⊢ (((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) ∧ (∀𝑘 ∈ 𝑥 𝜑 ∧ ∀𝑘 ∈ 𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
26	25	an4s 673	. . . . . . 7 ⊢ (((𝑥 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑥 𝜑) ∧ (𝑦 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
27	26	rexlimdvaa 3166	. . . . . 6 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑥 𝜑) → (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓)))
28	27	rexlimiva 3157	. . . . 5 ⊢ (∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 → (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓)))
29	28	imp 412	. . . 4 ⊢ ((∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
30		raleq 3318	. . . . . 6 ⊢ (𝑧 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)))
31	30	rexrn 7084	. . . . 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 221	. . 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 611	. 2 ⊢ ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
35	4, 34	impbii 212	1 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560  ran crn 5660   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  ℤcz 12619  ℤ_≥cuz 12891

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-pre-lttri 11202  ax-pre-lttrn 11203

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-po 5567  df-so 5568  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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-ov 7420  df-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-neg 11472  df-z 12620  df-uz 12892

This theorem is used by:  rexfiuz  15439  rexuz3  15440  rexanuz2  15441