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Theorem bnj601 32306
Description: Technical lemma for bnj852 32307. 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
bnj601.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj601.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj601.3 𝐷 = (ω ∖ {∅})
bnj601.4 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj601.5 (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))
Assertion
Ref Expression
bnj601 (𝑛 ≠ 1o → ((𝑛𝐷𝜃) → 𝜒))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑥,𝑓,𝑚,𝑛   𝜑,𝑖,𝑚   𝜓,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜒(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜃(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑥)   𝐷(𝑥,𝑦,𝑚,𝑛)   𝑅(𝑥)

Proof of Theorem bnj601
Dummy variables 𝑝 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj601.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2 bnj601.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj601.3 . 2 𝐷 = (ω ∖ {∅})
4 bnj601.4 . 2 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
5 bnj601.5 . 2 (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))
6 biid 264 . 2 ([𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜑)
7 biid 264 . 2 ([𝑚 / 𝑛]𝜓[𝑚 / 𝑛]𝜓)
8 biid 264 . 2 ([𝑚 / 𝑛]𝜒[𝑚 / 𝑛]𝜒)
9 bnj602 32301 . . . . . . 7 (𝑦 = 𝑧 → pred(𝑦, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
109cbviunv 4930 . . . . . 6 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)
1110opeq2i 4772 . . . . 5 𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩ = ⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩
1211sneqi 4539 . . . 4 {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}
1312uneq2i 4090 . . 3 (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})
14 dfsbcq 3725 . . 3 ((𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) → ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜑[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜑))
1513, 14ax-mp 5 . 2 ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜑[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜑)
16 dfsbcq 3725 . . 3 ((𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) → ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜓[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜓))
1713, 16ax-mp 5 . 2 ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜓[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜓)
18 dfsbcq 3725 . . 3 ((𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) → ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜒[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜒))
1913, 18ax-mp 5 . 2 ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜒[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜒)
2013eqcomi 2810 . 2 (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
21 biid 264 . 2 ((𝑓 Fn 𝑚[𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜓) ↔ (𝑓 Fn 𝑚[𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜓))
22 biid 264 . 2 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
23 biid 264 . 2 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
24 biid 264 . 2 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
25 biid 264 . 2 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
26 eqid 2801 . 2 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
27 eqid 2801 . 2 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
28 eqid 2801 . 2 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑖) pred(𝑦, 𝐴, 𝑅)
29 eqid 2801 . 2 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑝) pred(𝑦, 𝐴, 𝑅)
301, 2, 3, 4, 5, 6, 7, 8, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 20bnj600 32305 1 (𝑛 ≠ 1o → ((𝑛𝐷𝜃) → 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  ∃!weu 2631  wne 2990  wral 3109  [wsbc 3723  cdif 3881  cun 3882  c0 4246  {csn 4528  cop 4534   ciun 4884   class class class wbr 5033   E cep 5432  suc csuc 6165   Fn wfn 6323  cfv 6328  ωcom 7564  1oc1o 8082  w-bnj17 32070   predc-bnj14 32072   FrSe w-bnj15 32076
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-reg 9044
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1o 8089  df-bnj17 32071  df-bnj14 32073  df-bnj13 32075  df-bnj15 32077
This theorem is referenced by:  bnj852  32307
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