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Theorem bnj601 30244
 Description: Technical lemma for bnj852 30245. 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 250 . 2 ([𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜑)
7 biid 250 . 2 ([𝑚 / 𝑛]𝜓[𝑚 / 𝑛]𝜓)
8 biid 250 . 2 ([𝑚 / 𝑛]𝜒[𝑚 / 𝑛]𝜒)
9 bnj602 30239 . . . . . . 7 (𝑦 = 𝑧 → pred(𝑦, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
109cbviunv 4495 . . . . . 6 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)
1110opeq2i 4344 . . . . 5 𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩ = ⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩
1211sneqi 4136 . . . 4 {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}
1312uneq2i 3726 . . 3 (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})
14 dfsbcq 3404 . . 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 3404 . . 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 3404 . . 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 2619 . 2 (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
21 biid 250 . 2 ((𝑓 Fn 𝑚[𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜓) ↔ (𝑓 Fn 𝑚[𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜓))
22 biid 250 . 2 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
23 biid 250 . 2 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
24 biid 250 . 2 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
25 biid 250 . 2 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
26 eqid 2610 . 2 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
27 eqid 2610 . 2 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
28 eqid 2610 . 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 2610 . 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 30243 1 (𝑛 ≠ 1𝑜 → ((𝑛𝐷𝜃) → 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  ∀wral 2896  [wsbc 3402   ∖ cdif 3537   ∪ cun 3538  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583   E cep 4947  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957  1𝑜c1o 7440   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012 This theorem is referenced by:  bnj852  30245
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