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Theorem rexanuz 13876
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 3042 . . . 4 (∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
21rexbii 3019 . . 3 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ ∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
3 r19.40 3065 . . 3 (∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
42, 3sylbi 205 . 2 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
5 uzf 11519 . . . 4 :ℤ⟶𝒫 ℤ
6 ffn 5941 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
7 raleq 3111 . . . . 5 (𝑥 = (ℤ𝑗) → (∀𝑘𝑥 𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜑))
87rexrn 6251 . . . 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 3111 . . . . 5 (𝑦 = (ℤ𝑗) → (∀𝑘𝑦 𝜓 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜓))
1110rexrn 6251 . . . 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 13875 . . . . . . . . 9 ((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) → (𝑥𝑦) ∈ ran ℤ)
14 inss1 3791 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑥
15 ssralv 3625 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑥 → (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑))
1614, 15ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑)
17 inss2 3792 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑦
18 ssralv 3625 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑦 → (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓))
1917, 18ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓)
2016, 19anim12i 587 . . . . . . . . . 10 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
21 r19.26 3042 . . . . . . . . . 10 (∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓) ↔ (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
2220, 21sylibr 222 . . . . . . . . 9 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓))
23 raleq 3111 . . . . . . . . . 10 (𝑧 = (𝑥𝑦) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)))
2423rspcev 3278 . . . . . . . . 9 (((𝑥𝑦) ∈ ran ℤ ∧ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2513, 22, 24syl2an 492 . . . . . . . 8 (((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) ∧ (∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2625an4s 864 . . . . . . 7 (((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) ∧ (𝑦 ∈ ran ℤ ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2726rexlimdvaa 3010 . . . . . 6 ((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2827rexlimiva 3006 . . . . 5 (∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2928imp 443 . . . 4 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
30 raleq 3111 . . . . . 6 (𝑧 = (ℤ𝑗) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓)))
3130rexrn 6251 . . . . 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 206 . . 3 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
349, 12, 33syl2anbr 495 . 2 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
354, 34impbii 197 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 194  wa 382  wcel 1976  wral 2892  wrex 2893  cin 3535  wss 3536  𝒫 cpw 4104  ran crn 5026   Fn wfn 5782  wf 5783  cfv 5787  cz 11207  cuz 11516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-pre-lttri 9863  ax-pre-lttrn 9864
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-po 4946  df-so 4947  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-neg 10117  df-z 11208  df-uz 11517
This theorem is referenced by:  rexfiuz  13878  rexuz3  13879  rexanuz2  13880
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