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Theorem bnj544 34908
Description: Technical lemma for bnj852 34935. 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
bnj544.1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj544.2 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj544.3 𝐷 = (ω ∖ {∅})
bnj544.4 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj544.5 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj544.6 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
Assertion
Ref Expression
bnj544 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′
Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑚,𝑛)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑥,𝑓,𝑚,𝑛)   𝐺(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj544
StepHypRef Expression
1 bnj544.6 . . 3 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
2 bnj544.3 . . . . 5 𝐷 = (ω ∖ {∅})
32bnj923 34782 . . . 4 (𝑚𝐷𝑚 ∈ ω)
433anim1i 1153 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
51, 4sylbi 217 . 2 (𝜎 → (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
6 bnj544.1 . . 3 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
7 bnj544.2 . . 3 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
8 bnj544.4 . . 3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
9 bnj544.5 . . 3 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
10 biid 261 . . 3 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚) ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
116, 7, 8, 9, 10bnj543 34907 . 2 ((𝑅 FrSe 𝐴𝜏 ∧ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚)) → 𝐺 Fn 𝑛)
125, 11syl3an3 1166 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cun 3949  c0 4333  {csn 4626  cop 4632   ciun 4991  suc csuc 6386   Fn wfn 6556  cfv 6561  ωcom 7887   predc-bnj14 34702   FrSe w-bnj15 34706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-om 7888  df-bnj17 34701  df-bnj14 34703  df-bnj13 34705  df-bnj15 34707
This theorem is referenced by:  bnj600  34933  bnj908  34945
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