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Theorem bnj553 30942
Description: Technical lemma for bnj852 30965. 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 30814 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
3 opex 4923 . . . . . . 7 𝑚, 𝐶⟩ ∈ V
43snid 4199 . . . . . 6 𝑚, 𝐶⟩ ∈ {⟨𝑚, 𝐶⟩}
5 elun2 3773 . . . . . 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 2698 . . . 4 𝑚, 𝐶⟩ ∈ 𝐺
9 funopfv 6222 . . . 4 (Fun 𝐺 → (⟨𝑚, 𝐶⟩ ∈ 𝐺 → (𝐺𝑚) = 𝐶))
102, 8, 9mpisyl 21 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝐺𝑚) = 𝐶)
11103ad2ant1 1080 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐶)
12 fveq2 6178 . . . . . 6 (𝑝 = 𝑖 → (𝐺𝑝) = (𝐺𝑖))
1312bnj1113 30830 . . . . 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 2679 . . . 4 (𝑝 = 𝑖𝐿 = 𝐾)
17163ad2ant3 1082 . . 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 30941 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝐵 = 𝐾)
22213adant3 1079 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐵 = 𝐾)
23 fveq2 6178 . . . . . 6 (𝑝 = 𝑖 → (𝑓𝑝) = (𝑓𝑖))
2423bnj1113 30830 . . . . 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 2634 . . . . . 6 (𝐵 = 𝐶 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅))
27 eqcom 2627 . . . . . 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 1082 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐵 = 𝐶)
3117, 22, 303eqtr2rd 2661 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → 𝐶 = 𝐿)
3211, 31eqtrd 2654 1 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1481  wcel 1988  wral 2909  cdif 3564  cun 3565  c0 3907  {csn 4168  cop 4174   ciun 4511  suc csuc 5713  Fun wfun 5870   Fn wfn 5871  cfv 5876  ωcom 7050   predc-bnj14 30728   FrSe w-bnj15 30732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-res 5116  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884
This theorem is referenced by:  bnj557  30945
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