Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexfiuz Structured version   Visualization version   GIF version

Theorem rexfiuz 13935
 Description: Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)
Assertion
Ref Expression
rexfiuz (𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
Distinct variable groups:   𝑗,𝑘,𝑛,𝐴   𝜑,𝑗
Allowed substitution hints:   𝜑(𝑘,𝑛)

Proof of Theorem rexfiuz
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3115 . . . 4 (𝑥 = ∅ → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ ∅ 𝜑))
21rexralbidv 3040 . . 3 (𝑥 = ∅ → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑))
3 raleq 3115 . . 3 (𝑥 = ∅ → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
42, 3bibi12d 334 . 2 (𝑥 = ∅ → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
5 raleq 3115 . . . 4 (𝑥 = 𝑦 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑦 𝜑))
65rexralbidv 3040 . . 3 (𝑥 = 𝑦 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑))
7 raleq 3115 . . 3 (𝑥 = 𝑦 → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
86, 7bibi12d 334 . 2 (𝑥 = 𝑦 → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
9 raleq 3115 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
109rexralbidv 3040 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
11 raleq 3115 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
1210, 11bibi12d 334 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
13 raleq 3115 . . . 4 (𝑥 = 𝐴 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝐴 𝜑))
1413rexralbidv 3040 . . 3 (𝑥 = 𝐴 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑))
15 raleq 3115 . . 3 (𝑥 = 𝐴 → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
1614, 15bibi12d 334 . 2 (𝑥 = 𝐴 → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
17 0z 11265 . . . . 5 0 ∈ ℤ
1817ne0ii 3882 . . . 4 ℤ ≠ ∅
19 ral0 4028 . . . . 5 𝑛 ∈ ∅ 𝜑
2019rgen2w 2909 . . . 4 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
21 r19.2z 4012 . . . 4 ((ℤ ≠ ∅ ∧ ∀𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑)
2218, 20, 21mp2an 704 . . 3 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
23 ral0 4028 . . 3 𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑
2422, 232th 253 . 2 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)
25 anbi1 739 . . . 4 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
26 rexanuz 13933 . . . . 5 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
27 ralunb 3756 . . . . . . 7 (∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2827ralbii 2963 . . . . . 6 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2928rexbii 3023 . . . . 5 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
30 vex 3176 . . . . . . 7 𝑧 ∈ V
31 ralsnsg 4163 . . . . . . . 8 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑[𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
32 ralcom 3079 . . . . . . . . . . 11 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ𝑗)𝜑)
33 ralsnsg 4163 . . . . . . . . . . 11 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ𝑗)𝜑[𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
3432, 33syl5bb 271 . . . . . . . . . 10 (𝑧 ∈ V → (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑[𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
3534rexbidv 3034 . . . . . . . . 9 (𝑧 ∈ V → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
36 sbcrex 3481 . . . . . . . . 9 ([𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑)
3735, 36syl6rbbr 278 . . . . . . . 8 (𝑧 ∈ V → ([𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
3831, 37bitrd 267 . . . . . . 7 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
3930, 38ax-mp 5 . . . . . 6 (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑)
4039anbi2i 726 . . . . 5 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
4126, 29, 403bitr4i 291 . . . 4 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
42 ralunb 3756 . . . 4 (∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ (∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
4325, 41, 423bitr4g 302 . . 3 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
4443a1i 11 . 2 (𝑦 ∈ Fin → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
454, 8, 12, 16, 24, 44findcard2 8085 1 (𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173  [wsbc 3402   ∪ cun 3538  ∅c0 3874  {csn 4125  ‘cfv 5804  Fincfn 7841  0cc0 9815  ℤcz 11254  ℤ≥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-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  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:  uniioombllem6  23162  rrncmslem  32801
 Copyright terms: Public domain W3C validator