Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj865 Structured version   Visualization version   GIF version

Theorem bnj865 33929
Description: Technical lemma for bnj69 34016. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj865.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj865.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj865.3 𝐷 = (ω ∖ {∅})
bnj865.5 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
bnj865.6 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
Assertion
Ref Expression
bnj865 𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝑤,𝐴,𝑓,𝑛   𝐷,𝑓,𝑖,𝑛   𝑤,𝐷   𝑅,𝑓,𝑖,𝑛,𝑦   𝑤,𝑅   𝑓,𝑋,𝑛,𝑤   𝜑,𝑤   𝜓,𝑤
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝜒(𝑦,𝑤,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑤,𝑓,𝑖,𝑛)   𝐷(𝑦)   𝑋(𝑦,𝑖)

Proof of Theorem bnj865
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj865.1 . . . . . . 7 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj865.2 . . . . . . 7 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj865.3 . . . . . . 7 𝐷 = (ω ∖ {∅})
41, 2, 3bnj852 33927 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
5 omex 9637 . . . . . . . . 9 ω ∈ V
6 difexg 5327 . . . . . . . . 9 (ω ∈ V → (ω ∖ {∅}) ∈ V)
75, 6ax-mp 5 . . . . . . . 8 (ω ∖ {∅}) ∈ V
83, 7eqeltri 2829 . . . . . . 7 𝐷 ∈ V
9 raleq 3322 . . . . . . . 8 (𝑧 = 𝐷 → (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
10 raleq 3322 . . . . . . . . 9 (𝑧 = 𝐷 → (∀𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1110exbidv 1924 . . . . . . . 8 (𝑧 = 𝐷 → (∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
129, 11imbi12d 344 . . . . . . 7 (𝑧 = 𝐷 → ((∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
13 zfrep6 7940 . . . . . . 7 (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
148, 12, 13vtocl 3549 . . . . . 6 (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
154, 14syl 17 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
16 19.37v 1995 . . . . 5 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1715, 16mpbir 230 . . . 4 𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
18 df-ral 3062 . . . . . . . 8 (∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1918imbi2i 335 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
20 19.21v 1942 . . . . . . 7 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2119, 20bitr4i 277 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2221exbii 1850 . . . . 5 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
23 impexp 451 . . . . . . . 8 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
24 df-3an 1089 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
2524bicomi 223 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
2625imbi1i 349 . . . . . . . 8 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2723, 26bitr3i 276 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2827albii 1821 . . . . . 6 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2928exbii 1850 . . . . 5 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3022, 29bitri 274 . . . 4 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3117, 30mpbi 229 . . 3 𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
32 bnj865.5 . . . . . . 7 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
3332bicomi 223 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ 𝜒)
3433imbi1i 349 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3534albii 1821 . . . 4 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3635exbii 1850 . . 3 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3731, 36mpbi 229 . 2 𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
38 bnj865.6 . . . . . 6 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
3938rexbii 3094 . . . . 5 (∃𝑓𝑤 𝜃 ↔ ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
4039imbi2i 335 . . . 4 ((𝜒 → ∃𝑓𝑤 𝜃) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4140albii 1821 . . 3 (∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4241exbii 1850 . 2 (∃𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4337, 42mpbir 230 1 𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2562  wral 3061  wrex 3070  Vcvv 3474  cdif 3945  c0 4322  {csn 4628   ciun 4997  suc csuc 6366   Fn wfn 6538  cfv 6543  ωcom 7854   predc-bnj14 33694   FrSe w-bnj15 33698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-reg 9586  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-bnj17 33693  df-bnj14 33695  df-bnj13 33697  df-bnj15 33699
This theorem is referenced by:  bnj849  33931
  Copyright terms: Public domain W3C validator