Theorem rexanuz 15493

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 3123	. . . 4 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
2	1	rexbii 3110	. . 3 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
3		r19.40 3129	. . 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 12949	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
6		ffn 6701	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
7		raleq 3317	. . . . 5 ⊢ (𝑥 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑥 𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
8	7	rexrn 7079	. . . 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 3317	. . . . 5 ⊢ (𝑦 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑦 𝜓 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
11	10	rexrn 7079	. . . 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 15492	. . . . . . . . 9 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (𝑥 ∩ 𝑦) ∈ ran ℤ≥)
14		inss1 4182	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
15		ssralv 4000	. . . . . . . . . . . 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 4183	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
18		ssralv 4000	. . . . . . . . . . . 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 3123	. . . . . . . . . 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 3317	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓)))
24	23	rspcev 3577	. . . . . . . . 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 3165	. . . . . 6 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑥 𝜑) → (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓)))
28	27	rexlimiva 3156	. . . . 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 3317	. . . . . 6 ⊢ (𝑧 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)))
31	30	rexrn 7079	. . . . 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 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ran crn 5652   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  ℤcz 12674  ℤ_≥cuz 12946

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-pre-lttri 11255  ax-pre-lttrn 11256

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-neg 11525  df-z 12675  df-uz 12947

This theorem is used by:  rexfiuz  15495  rexuz3  15496  rexanuz2  15497