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Theorem rexanuz 14798
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 3085 . . . 4 (∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
21rexbii 3162 . . 3 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ ∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
3 r19.40 3251 . . 3 (∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
42, 3sylbi 220 . 2 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
5 uzf 12330 . . . 4 :ℤ⟶𝒫 ℤ
6 ffn 6505 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
7 raleq 3311 . . . . 5 (𝑥 = (ℤ𝑗) → (∀𝑘𝑥 𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜑))
87rexrn 6866 . . . 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 3311 . . . . 5 (𝑦 = (ℤ𝑗) → (∀𝑘𝑦 𝜓 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜓))
1110rexrn 6866 . . . 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 14797 . . . . . . . . 9 ((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) → (𝑥𝑦) ∈ ran ℤ)
14 inss1 4120 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑥
15 ssralv 3944 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑥 → (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑))
1614, 15ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑)
17 inss2 4121 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑦
18 ssralv 3944 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑦 → (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓))
1917, 18ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓)
2016, 19anim12i 616 . . . . . . . . . 10 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
21 r19.26 3085 . . . . . . . . . 10 (∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓) ↔ (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
2220, 21sylibr 237 . . . . . . . . 9 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓))
23 raleq 3311 . . . . . . . . . 10 (𝑧 = (𝑥𝑦) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)))
2423rspcev 3527 . . . . . . . . 9 (((𝑥𝑦) ∈ ran ℤ ∧ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2513, 22, 24syl2an 599 . . . . . . . 8 (((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) ∧ (∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2625an4s 660 . . . . . . 7 (((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) ∧ (𝑦 ∈ ran ℤ ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2726rexlimdvaa 3196 . . . . . 6 ((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2827rexlimiva 3192 . . . . 5 (∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2928imp 410 . . . 4 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
30 raleq 3311 . . . . . 6 (𝑧 = (ℤ𝑗) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓)))
3130rexrn 6866 . . . . 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 221 . . 3 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
349, 12, 33syl2anbr 602 . 2 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
354, 34impbii 212 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 209  wa 399  wcel 2114  wral 3054  wrex 3055  cin 3843  wss 3844  𝒫 cpw 4489  ran crn 5527   Fn wfn 6335  wf 6336  cfv 6340  cz 12065  cuz 12327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-cnex 10674  ax-resscn 10675  ax-pre-lttri 10692  ax-pre-lttrn 10693
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-po 5443  df-so 5444  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-er 8323  df-en 8559  df-dom 8560  df-sdom 8561  df-pnf 10758  df-mnf 10759  df-xr 10760  df-ltxr 10761  df-le 10762  df-neg 10954  df-z 12066  df-uz 12328
This theorem is referenced by:  rexfiuz  14800  rexuz3  14801  rexanuz2  14802
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