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Theorem rexanuz 13933
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 3046 . . . 4 (∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
21rexbii 3023 . . 3 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ ∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
3 r19.40 3069 . . 3 (∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
42, 3sylbi 206 . 2 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
5 uzf 11566 . . . 4 :ℤ⟶𝒫 ℤ
6 ffn 5958 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
7 raleq 3115 . . . . 5 (𝑥 = (ℤ𝑗) → (∀𝑘𝑥 𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜑))
87rexrn 6269 . . . 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 3115 . . . . 5 (𝑦 = (ℤ𝑗) → (∀𝑘𝑦 𝜓 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜓))
1110rexrn 6269 . . . 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 13932 . . . . . . . . 9 ((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) → (𝑥𝑦) ∈ ran ℤ)
14 inss1 3795 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑥
15 ssralv 3629 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑥 → (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑))
1614, 15ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑)
17 inss2 3796 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑦
18 ssralv 3629 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑦 → (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓))
1917, 18ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓)
2016, 19anim12i 588 . . . . . . . . . 10 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
21 r19.26 3046 . . . . . . . . . 10 (∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓) ↔ (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
2220, 21sylibr 223 . . . . . . . . 9 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓))
23 raleq 3115 . . . . . . . . . 10 (𝑧 = (𝑥𝑦) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)))
2423rspcev 3282 . . . . . . . . 9 (((𝑥𝑦) ∈ ran ℤ ∧ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2513, 22, 24syl2an 493 . . . . . . . 8 (((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) ∧ (∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2625an4s 865 . . . . . . 7 (((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) ∧ (𝑦 ∈ ran ℤ ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2726rexlimdvaa 3014 . . . . . 6 ((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2827rexlimiva 3010 . . . . 5 (∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2928imp 444 . . . 4 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
30 raleq 3115 . . . . . 6 (𝑧 = (ℤ𝑗) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓)))
3130rexrn 6269 . . . . 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 207 . . 3 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
349, 12, 33syl2anbr 496 . 2 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
354, 34impbii 198 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 195  wa 383  wcel 1977  wral 2896  wrex 2897  cin 3539  wss 3540  𝒫 cpw 4108  ran crn 5039   Fn wfn 5799  wf 5800  cfv 5804  cz 11254  cuz 11563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-neg 10148  df-z 11255  df-uz 11564
This theorem is referenced by:  rexfiuz  13935  rexuz3  13936  rexanuz2  13937
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