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Theorem bnj151 31275
Description: Technical lemma for bnj153 31278. 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 31274 . . . . . 6 𝜃0
1312, 6mpbi 220 . . . . 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 31267 . . . . . . 7 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2014, 15, 16, 17, 18, 19bnj149 31273 . . . . . 6 𝜃1
2120, 14mpbi 220 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′))
22 eu5 2633 . . . . 5 (∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′) ↔ (∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′) ∧ ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
2313, 21, 22sylanbrc 701 . . . 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 31272 . . . 4 (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
2723, 26mpbir 221 . . 3 𝜃′
28 sbceq1a 3587 . . . 4 (𝑛 = 1𝑜 → (𝜃[1𝑜 / 𝑛]𝜃))
2928, 25syl6bbr 278 . . 3 (𝑛 = 1𝑜 → (𝜃𝜃′))
3027, 29mpbiri 248 . 2 (𝑛 = 1𝑜𝜃)
3130a1d 25 1 (𝑛 = 1𝑜 → ((𝑛𝐷𝜏) → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  ∃!weu 2607  ∃*wmo 2608  wral 3050  [wsbc 3576  cdif 3712  c0 4058  {csn 4321  cop 4327   ciun 4672   class class class wbr 4804   E cep 5178  suc csuc 5886   Fn wfn 6044  cfv 6049  ωcom 7231  1𝑜c1o 7723   predc-bnj14 31084   FrSe w-bnj15 31088
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-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  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-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  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-1o 7730  df-bnj13 31087  df-bnj15 31089
This theorem is referenced by:  bnj153  31278
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