Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj864 Structured version   Visualization version   GIF version

Theorem bnj864 34558
Description: Technical lemma for bnj69 34646. 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
bnj864.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj864.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj864.3 𝐷 = (ω ∖ {∅})
bnj864.4 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
bnj864.5 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
Assertion
Ref Expression
bnj864 (𝜒 → ∃!𝑓𝜃)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑛
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑓,𝑖,𝑛)   𝐷(𝑦)   𝑋(𝑦,𝑖)

Proof of Theorem bnj864
StepHypRef Expression
1 bnj864.1 . . . . 5 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj864.2 . . . . 5 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj864.3 . . . . 5 𝐷 = (ω ∖ {∅})
41, 2, 3bnj852 34557 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
5 df-ral 3058 . . . . . 6 (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛(𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
65imbi2i 335 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))))
7 19.21v 1934 . . . . 5 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))))
8 impexp 449 . . . . . . 7 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))))
9 df-3an 1086 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
109bicomi 223 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
1110imbi1i 348 . . . . . . 7 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
128, 11bitr3i 276 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
1312albii 1813 . . . . 5 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
146, 7, 133bitr2i 298 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
154, 14mpbi 229 . . 3 𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
1615spi 2172 . 2 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
17 bnj864.4 . 2 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
18 bnj864.5 . . 3 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
1918eubii 2574 . 2 (∃!𝑓𝜃 ↔ ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
2016, 17, 193imtr4i 291 1 (𝜒 → ∃!𝑓𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wal 1531   = wceq 1533  wcel 2098  ∃!weu 2557  wral 3057  cdif 3944  c0 4324  {csn 4630   ciun 4998  suc csuc 6374   Fn wfn 6546  cfv 6551  ωcom 7874   predc-bnj14 34324   FrSe w-bnj15 34328
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-reg 9621  ax-inf2 9670
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-om 7875  df-1o 8491  df-bnj17 34323  df-bnj14 34325  df-bnj13 34327  df-bnj15 34329
This theorem is referenced by:  bnj849  34561
  Copyright terms: Public domain W3C validator