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Theorem bnj864 31299
Description: Technical lemma for bnj69 31385. 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 31298 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
5 df-ral 3055 . . . . . 6 (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛(𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
65imbi2i 325 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))))
7 19.21v 2017 . . . . 5 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))))
8 impexp 461 . . . . . . 7 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))))
9 df-3an 1074 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
109bicomi 214 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
1110imbi1i 338 . . . . . . 7 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
128, 11bitr3i 266 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
1312albii 1896 . . . . 5 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
146, 7, 133bitr2i 288 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
154, 14mpbi 220 . . 3 𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
1615spi 2201 . 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 2629 . 2 (∃!𝑓𝜃 ↔ ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
2016, 17, 193imtr4i 281 1 (𝜒 → ∃!𝑓𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wal 1630   = wceq 1632  wcel 2139  ∃!weu 2607  wral 3050  cdif 3712  c0 4058  {csn 4321   ciun 4672  suc csuc 5886   Fn wfn 6044  cfv 6049  ωcom 7230   predc-bnj14 31063   FrSe w-bnj15 31067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-reg 8662  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1o 7729  df-bnj17 31062  df-bnj14 31064  df-bnj13 31066  df-bnj15 31068
This theorem is referenced by:  bnj849  31302
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