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Theorem rexanuz 15289
Description: Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
Assertion
Ref Expression
rexanuz (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)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.
StepHypRef Expression
1 r19.26 3103 . . . 4 (∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
21rexbii 3086 . . 3 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ ∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
3 r19.40 3111 . . 3 (∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
42, 3sylbi 216 . 2 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
5 uzf 12822 . . . 4 :ℤ⟶𝒫 ℤ
6 ffn 6707 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
7 raleq 3314 . . . . 5 (𝑥 = (ℤ𝑗) → (∀𝑘𝑥 𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜑))
87rexrn 7078 . . . 4 (ℤ Fn ℤ → (∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
95, 6, 8mp2b 10 . . 3 (∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)
10 raleq 3314 . . . . 5 (𝑦 = (ℤ𝑗) → (∀𝑘𝑦 𝜓 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜓))
1110rexrn 7078 . . . 4 (ℤ Fn ℤ → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
125, 6, 11mp2b 10 . . 3 (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓)
13 uzin2 15288 . . . . . . . . 9 ((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) → (𝑥𝑦) ∈ ran ℤ)
14 inss1 4220 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑥
15 ssralv 4042 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑥 → (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑))
1614, 15ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑)
17 inss2 4221 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑦
18 ssralv 4042 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑦 → (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓))
1917, 18ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓)
2016, 19anim12i 612 . . . . . . . . . 10 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
21 r19.26 3103 . . . . . . . . . 10 (∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓) ↔ (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
2220, 21sylibr 233 . . . . . . . . 9 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓))
23 raleq 3314 . . . . . . . . . 10 (𝑧 = (𝑥𝑦) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)))
2423rspcev 3604 . . . . . . . . 9 (((𝑥𝑦) ∈ ran ℤ ∧ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2513, 22, 24syl2an 595 . . . . . . . 8 (((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) ∧ (∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2625an4s 657 . . . . . . 7 (((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) ∧ (𝑦 ∈ ran ℤ ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2726rexlimdvaa 3148 . . . . . 6 ((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2827rexlimiva 3139 . . . . 5 (∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2928imp 406 . . . 4 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
30 raleq 3314 . . . . . 6 (𝑧 = (ℤ𝑗) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓)))
3130rexrn 7078 . . . . 5 (ℤ Fn ℤ → (∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓)))
325, 6, 31mp2b 10 . . . 4 (∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
3329, 32sylib 217 . . 3 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
349, 12, 33syl2anbr 598 . 2 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
354, 34impbii 208 1 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2098  wral 3053  wrex 3062  cin 3939  wss 3940  𝒫 cpw 4594  ran crn 5667   Fn wfn 6528  wf 6529  cfv 6533  cz 12555  cuz 12819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-pre-lttri 11180  ax-pre-lttrn 11181
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-po 5578  df-so 5579  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-neg 11444  df-z 12556  df-uz 12820
This theorem is referenced by:  rexfiuz  15291  rexuz3  15292  rexanuz2  15293
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