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Theorem bnj151 32223
 Description: Technical lemma for bnj153 32226. 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
bnj151.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj151.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj151.3 𝐷 = (ω ∖ {∅})
bnj151.4 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj151.5 (𝜏 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜃))
bnj151.6 (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
bnj151.7 (𝜑′[1o / 𝑛]𝜑)
bnj151.8 (𝜓′[1o / 𝑛]𝜓)
bnj151.9 (𝜃′[1o / 𝑛]𝜃)
bnj151.10 (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1o𝜑′𝜓′)))
bnj151.11 (𝜃1 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1o𝜑′𝜓′)))
bnj151.12 (𝜁′[1o / 𝑛]𝜁)
bnj151.13 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
bnj151.14 (𝜑″[𝐹 / 𝑓]𝜑′)
bnj151.15 (𝜓″[𝐹 / 𝑓]𝜓′)
bnj151.16 (𝜁″[𝐹 / 𝑓]𝜁′)
bnj151.17 (𝜁0 ↔ (𝑓 Fn 1o𝜑′𝜓′))
bnj151.18 (𝜁1[𝑔 / 𝑓]𝜁0)
bnj151.19 (𝜑1[𝑔 / 𝑓]𝜑′)
bnj151.20 (𝜓1[𝑔 / 𝑓]𝜓′)
Assertion
Ref Expression
bnj151 (𝑛 = 1o → ((𝑛𝐷𝜏) → 𝜃))
Distinct variable groups:   𝐴,𝑓,𝑔,𝑥   𝐴,𝑛,𝑓,𝑥   𝑓,𝐹,𝑖,𝑦   𝑅,𝑓,𝑔,𝑥   𝑅,𝑛   𝑓,𝜁1   𝑓,𝜁″   𝑔,𝜁0   𝑖,𝑛,𝑦   𝑚,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜃(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜏(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜁(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝐴(𝑦,𝑖,𝑚)   𝐷(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝑅(𝑦,𝑖,𝑚)   𝐹(𝑥,𝑔,𝑚,𝑛)   𝜑′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜃′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜁′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜑″(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜓″(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜁″(𝑥,𝑦,𝑔,𝑖,𝑚,𝑛)   𝜃0(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜁0(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜑1(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜓1(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜃1(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛)   𝜁1(𝑥,𝑦,𝑔,𝑖,𝑚,𝑛)

Proof of Theorem bnj151
StepHypRef Expression
1 bnj151.1 . . . . . . 7 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2 bnj151.2 . . . . . . 7 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj151.6 . . . . . . 7 (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
4 bnj151.7 . . . . . . 7 (𝜑′[1o / 𝑛]𝜑)
5 bnj151.8 . . . . . . 7 (𝜓′[1o / 𝑛]𝜓)
6 bnj151.10 . . . . . . 7 (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1o𝜑′𝜓′)))
7 bnj151.12 . . . . . . 7 (𝜁′[1o / 𝑛]𝜁)
8 bnj151.13 . . . . . . 7 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
9 bnj151.14 . . . . . . 7 (𝜑″[𝐹 / 𝑓]𝜑′)
10 bnj151.15 . . . . . . 7 (𝜓″[𝐹 / 𝑓]𝜓′)
11 bnj151.16 . . . . . . 7 (𝜁″[𝐹 / 𝑓]𝜁′)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj150 32222 . . . . . 6 𝜃0
1312, 6mpbi 233 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1o𝜑′𝜓′))
14 bnj151.11 . . . . . . 7 (𝜃1 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1o𝜑′𝜓′)))
15 bnj151.17 . . . . . . 7 (𝜁0 ↔ (𝑓 Fn 1o𝜑′𝜓′))
16 bnj151.18 . . . . . . 7 (𝜁1[𝑔 / 𝑓]𝜁0)
17 bnj151.19 . . . . . . 7 (𝜑1[𝑔 / 𝑓]𝜑′)
18 bnj151.20 . . . . . . 7 (𝜓1[𝑔 / 𝑓]𝜓′)
191, 4bnj118 32215 . . . . . . 7 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2014, 15, 16, 17, 18, 19bnj149 32221 . . . . . 6 𝜃1
2120, 14mpbi 233 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1o𝜑′𝜓′))
22 df-eu 2653 . . . . 5 (∃!𝑓(𝑓 Fn 1o𝜑′𝜓′) ↔ (∃𝑓(𝑓 Fn 1o𝜑′𝜓′) ∧ ∃*𝑓(𝑓 Fn 1o𝜑′𝜓′)))
2313, 21, 22sylanbrc 586 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1o𝜑′𝜓′))
24 bnj151.4 . . . . 5 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
25 bnj151.9 . . . . 5 (𝜃′[1o / 𝑛]𝜃)
2624, 4, 5, 25bnj130 32220 . . . 4 (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1o𝜑′𝜓′)))
2723, 26mpbir 234 . . 3 𝜃′
28 sbceq1a 3758 . . . 4 (𝑛 = 1o → (𝜃[1o / 𝑛]𝜃))
2928, 25syl6bbr 292 . . 3 (𝑛 = 1o → (𝜃𝜃′))
3027, 29mpbiri 261 . 2 (𝑛 = 1o𝜃)
3130a1d 25 1 (𝑛 = 1o → ((𝑛𝐷𝜏) → 𝜃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2620  ∃!weu 2652  ∀wral 3130  [wsbc 3747   ∖ cdif 3905  ∅c0 4265  {csn 4539  ⟨cop 4545  ∪ ciun 4894   class class class wbr 5042   E cep 5441  suc csuc 6171   Fn wfn 6329  ‘cfv 6334  ωcom 7565  1oc1o 8082   predc-bnj14 32032   FrSe w-bnj15 32036 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-reu 3137  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-1o 8089  df-bnj13 32035  df-bnj15 32037 This theorem is referenced by:  bnj153  32226
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