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Theorem bnj601 33826
Description: Technical lemma for bnj852 33827. 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 260 . 2 ([𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜑)
7 biid 260 . 2 ([𝑚 / 𝑛]𝜓[𝑚 / 𝑛]𝜓)
8 biid 260 . 2 ([𝑚 / 𝑛]𝜒[𝑚 / 𝑛]𝜒)
9 bnj602 33821 . . . . . . 7 (𝑦 = 𝑧 → pred(𝑦, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
109cbviunv 5037 . . . . . 6 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)
1110opeq2i 4871 . . . . 5 𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩ = ⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩
1211sneqi 4634 . . . 4 {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}
1312uneq2i 4157 . . 3 (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})
14 dfsbcq 3776 . . 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 3776 . . 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 3776 . . 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 2741 . 2 (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
21 biid 260 . 2 ((𝑓 Fn 𝑚[𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜓) ↔ (𝑓 Fn 𝑚[𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜓))
22 biid 260 . 2 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
23 biid 260 . 2 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
24 biid 260 . 2 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
25 biid 260 . 2 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
26 eqid 2732 . 2 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
27 eqid 2732 . 2 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
28 eqid 2732 . 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 2732 . 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 33825 1 (𝑛 ≠ 1o → ((𝑛𝐷𝜃) → 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  ∃!weu 2562  wne 2940  wral 3061  [wsbc 3774  cdif 3942  cun 3943  c0 4319  {csn 4623  cop 4629   ciun 4991   class class class wbr 5142   E cep 5573  suc csuc 6356   Fn wfn 6528  cfv 6533  ωcom 7839  1oc1o 8443  w-bnj17 33592   predc-bnj14 33594   FrSe w-bnj15 33598
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709  ax-reg 9571
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-fv 6541  df-om 7840  df-1o 8450  df-bnj17 33593  df-bnj14 33595  df-bnj13 33597  df-bnj15 33599
This theorem is referenced by:  bnj852  33827
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