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Theorem bnj553 32165
Description: Technical lemma for bnj852 32188. 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 32036 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
3 opex 5349 . . . . . . 7 𝑚, 𝐶⟩ ∈ V
43snid 4595 . . . . . 6 𝑚, 𝐶⟩ ∈ {⟨𝑚, 𝐶⟩}
5 elun2 4153 . . . . . 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 2912 . . . 4 𝑚, 𝐶⟩ ∈ 𝐺
9 funopfv 6712 . . . 4 (Fun 𝐺 → (⟨𝑚, 𝐶⟩ ∈ 𝐺 → (𝐺𝑚) = 𝐶))
102, 8, 9mpisyl 21 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝐺𝑚) = 𝐶)
11103ad2ant1 1129 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐶)
12 fveq2 6665 . . . . . 6 (𝑝 = 𝑖 → (𝐺𝑝) = (𝐺𝑖))
1312bnj1113 32052 . . . . 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 2881 . . . 4 (𝑝 = 𝑖𝐿 = 𝐾)
17163ad2ant3 1131 . . 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 32164 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝐵 = 𝐾)
22213adant3 1128 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐵 = 𝐾)
23 fveq2 6665 . . . . . 6 (𝑝 = 𝑖 → (𝑓𝑝) = (𝑓𝑖))
2423bnj1113 32052 . . . . 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 2836 . . . . . 6 (𝐵 = 𝐶 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅))
27 eqcom 2828 . . . . . 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 277 . . . . 5 (𝐵 = 𝐶 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2924, 28sylibr 236 . . . 4 (𝑝 = 𝑖𝐵 = 𝐶)
30293ad2ant3 1131 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐵 = 𝐶)
3117, 22, 303eqtr2rd 2863 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐶 = 𝐿)
3211, 31eqtrd 2856 1 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  w3a 1083   = wceq 1533  wcel 2110  wral 3138  cdif 3933  cun 3934  c0 4291  {csn 4561  cop 4567   ciun 4912  suc csuc 6188  Fun wfun 6344   Fn wfn 6345  cfv 6350  ωcom 7574   predc-bnj14 31953   FrSe w-bnj15 31957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-res 5562  df-iota 6309  df-fun 6352  df-fn 6353  df-fv 6358
This theorem is referenced by:  bnj557  32168
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