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Theorem bnj130 32153
Description: Technical lemma for bnj151 32156. 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
bnj130.1 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj130.2 (𝜑′[1o / 𝑛]𝜑)
bnj130.3 (𝜓′[1o / 𝑛]𝜓)
bnj130.4 (𝜃′[1o / 𝑛]𝜃)
Assertion
Ref Expression
bnj130 (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1o𝜑′𝜓′)))
Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑓,𝑛   𝑥,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝜓(𝑥,𝑓,𝑛)   𝜃(𝑥,𝑓,𝑛)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝜑′(𝑥,𝑓,𝑛)   𝜓′(𝑥,𝑓,𝑛)   𝜃′(𝑥,𝑓,𝑛)

Proof of Theorem bnj130
StepHypRef Expression
1 bnj130.1 . . 3 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
21sbcbii 3805 . 2 ([1o / 𝑛]𝜃[1o / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3 bnj130.4 . 2 (𝜃′[1o / 𝑛]𝜃)
4 bnj105 32001 . . . . . . . . . 10 1o ∈ V
54bnj90 31999 . . . . . . . . 9 ([1o / 𝑛]𝑓 Fn 𝑛𝑓 Fn 1o)
65bicomi 227 . . . . . . . 8 (𝑓 Fn 1o[1o / 𝑛]𝑓 Fn 𝑛)
7 bnj130.2 . . . . . . . 8 (𝜑′[1o / 𝑛]𝜑)
8 bnj130.3 . . . . . . . 8 (𝜓′[1o / 𝑛]𝜓)
96, 7, 83anbi123i 1152 . . . . . . 7 ((𝑓 Fn 1o𝜑′𝜓′) ↔ ([1o / 𝑛]𝑓 Fn 𝑛[1o / 𝑛]𝜑[1o / 𝑛]𝜓))
10 sbc3an 3814 . . . . . . 7 ([1o / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ ([1o / 𝑛]𝑓 Fn 𝑛[1o / 𝑛]𝜑[1o / 𝑛]𝜓))
119, 10bitr4i 281 . . . . . 6 ((𝑓 Fn 1o𝜑′𝜓′) ↔ [1o / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1211eubii 2670 . . . . 5 (∃!𝑓(𝑓 Fn 1o𝜑′𝜓′) ↔ ∃!𝑓[1o / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
134bnj89 31998 . . . . 5 ([1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓[1o / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1412, 13bitr4i 281 . . . 4 (∃!𝑓(𝑓 Fn 1o𝜑′𝜓′) ↔ [1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
1514imbi2i 339 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1o𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
16 nfv 1916 . . . . 5 𝑛(𝑅 FrSe 𝐴𝑥𝐴)
1716sbc19.21g 3821 . . . 4 (1o ∈ V → ([1o / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))))
184, 17ax-mp 5 . . 3 ([1o / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
1915, 18bitr4i 281 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1o𝜑′𝜓′)) ↔ [1o / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
202, 3, 193bitr4i 306 1 (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1o𝜑′𝜓′)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wcel 2115  ∃!weu 2653  Vcvv 3471  [wsbc 3749   Fn wfn 6323  1oc1o 8070   FrSe w-bnj15 31969
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-pw 4514  df-sn 4541  df-suc 6170  df-fn 6331  df-1o 8077
This theorem is referenced by:  bnj151  32156
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