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Theorem rexanuz 14462
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 3274 . . . 4 (∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
21rexbii 3251 . . 3 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ ∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓))
3 r19.40 3298 . . 3 (∃𝑗 ∈ ℤ (∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑘 ∈ (ℤ𝑗)𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
42, 3sylbi 209 . 2 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))
5 uzf 11971 . . . 4 :ℤ⟶𝒫 ℤ
6 ffn 6278 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
7 raleq 3350 . . . . 5 (𝑥 = (ℤ𝑗) → (∀𝑘𝑥 𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜑))
87rexrn 6610 . . . 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 3350 . . . . 5 (𝑦 = (ℤ𝑗) → (∀𝑘𝑦 𝜓 ↔ ∀𝑘 ∈ (ℤ𝑗)𝜓))
1110rexrn 6610 . . . 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 14461 . . . . . . . . 9 ((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) → (𝑥𝑦) ∈ ran ℤ)
14 inss1 4057 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑥
15 ssralv 3891 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑥 → (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑))
1614, 15ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑥 𝜑 → ∀𝑘 ∈ (𝑥𝑦)𝜑)
17 inss2 4058 . . . . . . . . . . . 12 (𝑥𝑦) ⊆ 𝑦
18 ssralv 3891 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝑦 → (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓))
1917, 18ax-mp 5 . . . . . . . . . . 11 (∀𝑘𝑦 𝜓 → ∀𝑘 ∈ (𝑥𝑦)𝜓)
2016, 19anim12i 608 . . . . . . . . . 10 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
21 r19.26 3274 . . . . . . . . . 10 (∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓) ↔ (∀𝑘 ∈ (𝑥𝑦)𝜑 ∧ ∀𝑘 ∈ (𝑥𝑦)𝜓))
2220, 21sylibr 226 . . . . . . . . 9 ((∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓) → ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓))
23 raleq 3350 . . . . . . . . . 10 (𝑧 = (𝑥𝑦) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)))
2423rspcev 3526 . . . . . . . . 9 (((𝑥𝑦) ∈ ran ℤ ∧ ∀𝑘 ∈ (𝑥𝑦)(𝜑𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2513, 22, 24syl2an 591 . . . . . . . 8 (((𝑥 ∈ ran ℤ𝑦 ∈ ran ℤ) ∧ (∀𝑘𝑥 𝜑 ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2625an4s 652 . . . . . . 7 (((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) ∧ (𝑦 ∈ ran ℤ ∧ ∀𝑘𝑦 𝜓)) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
2726rexlimdvaa 3241 . . . . . 6 ((𝑥 ∈ ran ℤ ∧ ∀𝑘𝑥 𝜑) → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2827rexlimiva 3237 . . . . 5 (∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 → (∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓 → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓)))
2928imp 397 . . . 4 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑧 ∈ ran ℤ𝑘𝑧 (𝜑𝜓))
30 raleq 3350 . . . . . 6 (𝑧 = (ℤ𝑗) → (∀𝑘𝑧 (𝜑𝜓) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓)))
3130rexrn 6610 . . . . 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 210 . . 3 ((∃𝑥 ∈ ran ℤ𝑘𝑥 𝜑 ∧ ∃𝑦 ∈ ran ℤ𝑘𝑦 𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
349, 12, 33syl2anbr 594 . 2 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
354, 34impbii 201 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 198  wa 386  wcel 2166  wral 3117  wrex 3118  cin 3797  wss 3798  𝒫 cpw 4378  ran crn 5343   Fn wfn 6118  wf 6119  cfv 6123  cz 11704  cuz 11968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-pre-lttri 10326  ax-pre-lttrn 10327
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-po 5263  df-so 5264  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-neg 10588  df-z 11705  df-uz 11969
This theorem is referenced by:  rexfiuz  14464  rexuz3  14465  rexanuz2  14466
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