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Theorem bnj553 31275
Description: Technical lemma for bnj852 31298. 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
bnj553.1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj553.2 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj553.3 𝐷 = (ω ∖ {∅})
bnj553.4 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj553.5 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj553.6 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj553.7 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
bnj553.8 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj553.9 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
bnj553.10 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj553.11 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
bnj553.12 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Assertion
Ref Expression
bnj553 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐿)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑦,𝐺   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′
Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑚,𝑛)   𝐵(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑥,𝑓,𝑚,𝑛)   𝐺(𝑥,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐾(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐿(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj553
StepHypRef Expression
1 bnj553.12 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
21bnj930 31147 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
3 opex 5081 . . . . . . 7 𝑚, 𝐶⟩ ∈ V
43snid 4353 . . . . . 6 𝑚, 𝐶⟩ ∈ {⟨𝑚, 𝐶⟩}
5 elun2 3924 . . . . . 6 (⟨𝑚, 𝐶⟩ ∈ {⟨𝑚, 𝐶⟩} → ⟨𝑚, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑚, 𝐶⟩}))
64, 5ax-mp 5 . . . . 5 𝑚, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑚, 𝐶⟩})
7 bnj553.8 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
86, 7eleqtrri 2838 . . . 4 𝑚, 𝐶⟩ ∈ 𝐺
9 funopfv 6396 . . . 4 (Fun 𝐺 → (⟨𝑚, 𝐶⟩ ∈ 𝐺 → (𝐺𝑚) = 𝐶))
102, 8, 9mpisyl 21 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝐺𝑚) = 𝐶)
11103ad2ant1 1128 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐶)
12 fveq2 6352 . . . . . 6 (𝑝 = 𝑖 → (𝐺𝑝) = (𝐺𝑖))
1312bnj1113 31163 . . . . 5 (𝑝 = 𝑖 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
14 bnj553.11 . . . . 5 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
15 bnj553.10 . . . . 5 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
1613, 14, 153eqtr4g 2819 . . . 4 (𝑝 = 𝑖𝐿 = 𝐾)
17163ad2ant3 1130 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐿 = 𝐾)
18 bnj553.5 . . . . 5 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
19 bnj553.9 . . . . 5 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
20 bnj553.4 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
2118, 19, 15, 20, 1bnj548 31274 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝐵 = 𝐾)
22213adant3 1127 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐵 = 𝐾)
23 fveq2 6352 . . . . . 6 (𝑝 = 𝑖 → (𝑓𝑝) = (𝑓𝑖))
2423bnj1113 31163 . . . . 5 (𝑝 = 𝑖 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
25 bnj553.7 . . . . . . 7 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
2619, 25eqeq12i 2774 . . . . . 6 (𝐵 = 𝐶 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅))
27 eqcom 2767 . . . . . 6 ( 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ↔ 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2826, 27bitri 264 . . . . 5 (𝐵 = 𝐶 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2924, 28sylibr 224 . . . 4 (𝑝 = 𝑖𝐵 = 𝐶)
30293ad2ant3 1130 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐵 = 𝐶)
3117, 22, 303eqtr2rd 2801 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐶 = 𝐿)
3211, 31eqtrd 2794 1 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1632  wcel 2139  wral 3050  cdif 3712  cun 3713  c0 4058  {csn 4321  cop 4327   ciun 4672  suc csuc 5886  Fun wfun 6043   Fn wfn 6044  cfv 6049  ωcom 7230   predc-bnj14 31063   FrSe w-bnj15 31067
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-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-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-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  bnj557  31278
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