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Theorem bnj151 31464
Description: Technical lemma for bnj153 31467. 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 31463 . . . . . 6 𝜃0
1312, 6mpbi 222 . . . . 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 31456 . . . . . . 7 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2014, 15, 16, 17, 18, 19bnj149 31462 . . . . . 6 𝜃1
2120, 14mpbi 222 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′))
22 df-eu 2609 . . . . 5 (∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′) ↔ (∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′) ∧ ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
2313, 21, 22sylanbrc 579 . . . 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 31461 . . . 4 (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
2723, 26mpbir 223 . . 3 𝜃′
28 sbceq1a 3644 . . . 4 (𝑛 = 1𝑜 → (𝜃[1𝑜 / 𝑛]𝜃))
2928, 25syl6bbr 281 . . 3 (𝑛 = 1𝑜 → (𝜃𝜃′))
3027, 29mpbiri 250 . 2 (𝑛 = 1𝑜𝜃)
3130a1d 25 1 (𝑛 = 1𝑜 → ((𝑛𝐷𝜏) → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  ∃*wmo 2589  ∃!weu 2608  wral 3089  [wsbc 3633  cdif 3766  c0 4115  {csn 4368  cop 4374   ciun 4710   class class class wbr 4843   E cep 5224  suc csuc 5943   Fn wfn 6096  cfv 6101  ωcom 7299  1𝑜c1o 7792   predc-bnj14 31274   FrSe w-bnj15 31278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-1o 7799  df-bnj13 31277  df-bnj15 31279
This theorem is referenced by:  bnj153  31467
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