Theorem rexanuz 11507

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 2657	. . . 4 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
2	1	rexbii 2537	. . 3 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
3		r19.40 2685	. . 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 121	. 2 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
5		uzf 9733	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
6		ffn 5473	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
7		raleq 2728	. . . . 5 ⊢ (𝑥 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑥 𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
8	7	rexrn 5774	. . . 4 ⊢ (ℤ≥ Fn ℤ → (∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
9	5, 6, 8	mp2b 8	. . 3 ⊢ (∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)
10		raleq 2728	. . . . 5 ⊢ (𝑦 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑦 𝜓 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
11	10	rexrn 5774	. . . 4 ⊢ (ℤ≥ Fn ℤ → (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
12	5, 6, 11	mp2b 8	. . 3 ⊢ (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)
13		uzin2 11506	. . . . . . . . 9 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) → (𝑥 ∩ 𝑦) ∈ ran ℤ≥)
14		inss1 3424	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
15		ssralv 3288	. . . . . . . . . . . 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 3425	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
18		ssralv 3288	. . . . . . . . . . . 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 338	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝑥 𝜑 ∧ ∀𝑘 ∈ 𝑦 𝜓) → (∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜓))
21		r19.26 2657	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓) ↔ (∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥 ∩ 𝑦)𝜓))
22	20, 21	sylibr 134	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ 𝑥 𝜑 ∧ ∀𝑘 ∈ 𝑦 𝜓) → ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓))
23		raleq 2728	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓)))
24	23	rspcev 2907	. . . . . . . . 9 ⊢ (((𝑥 ∩ 𝑦) ∈ ran ℤ≥ ∧ ∀𝑘 ∈ (𝑥 ∩ 𝑦)(𝜑 ∧ 𝜓)) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
25	13, 22, 24	syl2an 289	. . . . . . . 8 ⊢ (((𝑥 ∈ ran ℤ≥ ∧ 𝑦 ∈ ran ℤ≥) ∧ (∀𝑘 ∈ 𝑥 𝜑 ∧ ∀𝑘 ∈ 𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
26	25	an4s 590	. . . . . . 7 ⊢ (((𝑥 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑥 𝜑) ∧ (𝑦 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
27	26	rexlimdvaa 2649	. . . . . 6 ⊢ ((𝑥 ∈ ran ℤ≥ ∧ ∀𝑘 ∈ 𝑥 𝜑) → (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓)))
28	27	rexlimiva 2643	. . . . 5 ⊢ (∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 → (∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓 → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓)))
29	28	imp 124	. . . 4 ⊢ ((∃𝑥 ∈ ran ℤ≥∀𝑘 ∈ 𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ≥∀𝑘 ∈ 𝑦 𝜓) → ∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓))
30		raleq 2728	. . . . . 6 ⊢ (𝑧 = (ℤ≥‘𝑗) → (∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)))
31	30	rexrn 5774	. . . . 5 ⊢ (ℤ≥ Fn ℤ → (∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)))
32	5, 6, 31	mp2b 8	. . . 4 ⊢ (∃𝑧 ∈ ran ℤ≥∀𝑘 ∈ 𝑧 (𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
33	29, 32	sylib 122	. . 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 292	. 2 ⊢ ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
35	4, 34	impbii 126	1 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   ∩ cin 3196   ⊆ wss 3197  𝒫 cpw 3649  ran crn 4720   Fn wfn 5313  ⟶wf 5314  ‘cfv 5318  ℤcz 9454  ℤ_≥cuz 9730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-inn 9119  df-n0 9378  df-z 9455  df-uz 9731

This theorem is referenced by:  rexfiuz  11508  rexuz3  11509  rexanuz2  11510