Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj579 Structured version   Visualization version   GIF version

Theorem bnj579 34453
Description: Technical lemma for bnj852 34460. 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
bnj579.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj579.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj579.3 𝐷 = (ω ∖ {∅})
Assertion
Ref Expression
bnj579 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
Distinct variable groups:   𝐴,𝑓,𝑖   𝐷,𝑓   𝑅,𝑓,𝑖   𝑓,𝑛,𝑖   𝑥,𝑓   𝑦,𝑓,𝑖
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝐴(𝑥,𝑦,𝑛)   𝐷(𝑥,𝑦,𝑖,𝑛)   𝑅(𝑥,𝑦,𝑛)

Proof of Theorem bnj579
Dummy variables 𝑘 𝑔 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj579.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2 bnj579.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 biid 261 . 2 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
4 biid 261 . 2 ([𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜑)
5 biid 261 . 2 ([𝑔 / 𝑓]𝜓[𝑔 / 𝑓]𝜓)
6 biid 261 . 2 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓))
7 bnj579.3 . 2 𝐷 = (ω ∖ {∅})
8 biid 261 . 2 (((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗)))
9 biid 261 . 2 (∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗))) ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗))))
101, 2, 3, 4, 5, 6, 7, 8, 9bnj580 34452 1 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  ∃*wmo 2526  wral 3055  [wsbc 3772  cdif 3940  c0 4317  {csn 4623   ciun 4990   class class class wbr 5141   E cep 5572  suc csuc 6359   Fn wfn 6531  cfv 6536  ωcom 7851   predc-bnj14 34227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544  df-om 7852  df-bnj17 34226
This theorem is referenced by:  bnj600  34458
  Copyright terms: Public domain W3C validator