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Theorem bnj966 32924
Description: Technical lemma for bnj69 32990. 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
bnj966.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj966.10 𝐷 = (ω ∖ {∅})
bnj966.12 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj966.13 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj966.44 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
bnj966.53 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)
Assertion
Ref Expression
bnj966 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
Distinct variable groups:   𝑦,𝑓   𝑦,𝑖   𝑦,𝑚   𝑦,𝑛
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐺(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj966
StepHypRef Expression
1 bnj966.53 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)
21fnfund 6534 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → Fun 𝐺)
323adant3 1131 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → Fun 𝐺)
4 opex 5379 . . . . . . 7 𝑛, 𝐶⟩ ∈ V
54snid 4597 . . . . . 6 𝑛, 𝐶⟩ ∈ {⟨𝑛, 𝐶⟩}
6 elun2 4111 . . . . . 6 (⟨𝑛, 𝐶⟩ ∈ {⟨𝑛, 𝐶⟩} → ⟨𝑛, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑛, 𝐶⟩}))
75, 6ax-mp 5 . . . . 5 𝑛, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑛, 𝐶⟩})
8 bnj966.13 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
97, 8eleqtrri 2838 . . . 4 𝑛, 𝐶⟩ ∈ 𝐺
10 funopfv 6821 . . . 4 (Fun 𝐺 → (⟨𝑛, 𝐶⟩ ∈ 𝐺 → (𝐺𝑛) = 𝐶))
113, 9, 10mpisyl 21 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺𝑛) = 𝐶)
12 simp22 1206 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑛 = suc 𝑚)
13 simp33 1210 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑛 = suc 𝑖)
14 bnj551 32722 . . . . 5 ((𝑛 = suc 𝑚𝑛 = suc 𝑖) → 𝑚 = 𝑖)
1512, 13, 14syl2anc 584 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑚 = 𝑖)
16 suceq 6331 . . . . . . . 8 (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖)
1716eqeq2d 2749 . . . . . . 7 (𝑚 = 𝑖 → (𝑛 = suc 𝑚𝑛 = suc 𝑖))
1817biimpac 479 . . . . . 6 ((𝑛 = suc 𝑚𝑚 = 𝑖) → 𝑛 = suc 𝑖)
1918fveq2d 6778 . . . . 5 ((𝑛 = suc 𝑚𝑚 = 𝑖) → (𝐺𝑛) = (𝐺‘suc 𝑖))
20 bnj966.12 . . . . . . 7 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
21 fveq2 6774 . . . . . . . 8 (𝑚 = 𝑖 → (𝑓𝑚) = (𝑓𝑖))
2221bnj1113 32765 . . . . . . 7 (𝑚 = 𝑖 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2320, 22eqtrid 2790 . . . . . 6 (𝑚 = 𝑖𝐶 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2423adantl 482 . . . . 5 ((𝑛 = suc 𝑚𝑚 = 𝑖) → 𝐶 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2519, 24eqeq12d 2754 . . . 4 ((𝑛 = suc 𝑚𝑚 = 𝑖) → ((𝐺𝑛) = 𝐶 ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2612, 15, 25syl2anc 584 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → ((𝐺𝑛) = 𝐶 ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2711, 26mpbid 231 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
28 bnj966.44 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
29283adant3 1131 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝐶 ∈ V)
30 bnj966.3 . . . . . . . 8 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
3130bnj1235 32784 . . . . . . 7 (𝜒𝑓 Fn 𝑛)
32313ad2ant1 1132 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
33323ad2ant2 1133 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑓 Fn 𝑛)
34 simp23 1207 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑝 = suc 𝑛)
3529, 33, 34, 13bnj951 32755 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖))
36 bnj966.10 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
3736bnj923 32748 . . . . . . . 8 (𝑛𝐷𝑛 ∈ ω)
3830, 37bnj769 32742 . . . . . . 7 (𝜒𝑛 ∈ ω)
39383ad2ant1 1132 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛 ∈ ω)
40 simp3 1137 . . . . . 6 ((𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖) → 𝑛 = suc 𝑖)
4139, 40bnj240 32678 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝑛 ∈ ω ∧ 𝑛 = suc 𝑖))
42 vex 3436 . . . . . . 7 𝑖 ∈ V
4342bnj216 32711 . . . . . 6 (𝑛 = suc 𝑖𝑖𝑛)
4443adantl 482 . . . . 5 ((𝑛 ∈ ω ∧ 𝑛 = suc 𝑖) → 𝑖𝑛)
4541, 44syl 17 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑖𝑛)
46 bnj658 32731 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛))
4746anim1i 615 . . . . . 6 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) ∧ 𝑖𝑛) → ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
48 df-bnj17 32666 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
4947, 48sylibr 233 . . . . 5 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) ∧ 𝑖𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛))
508bnj945 32753 . . . . 5 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) → (𝐺𝑖) = (𝑓𝑖))
5149, 50syl 17 . . . 4 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) ∧ 𝑖𝑛) → (𝐺𝑖) = (𝑓𝑖))
5235, 45, 51syl2anc 584 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺𝑖) = (𝑓𝑖))
5320, 8bnj958 32920 . . . . 5 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
5453bnj956 32756 . . . 4 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
5554eqeq2d 2749 . . 3 ((𝐺𝑖) = (𝑓𝑖) → ((𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5652, 55syl 17 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → ((𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5727, 56mpbird 256 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  cdif 3884  cun 3885  c0 4256  {csn 4561  cop 4567   ciun 4924  suc csuc 6268  Fun wfun 6427   Fn wfn 6428  cfv 6433  ωcom 7712  w-bnj17 32665   predc-bnj14 32667   FrSe w-bnj15 32671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-reg 9351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-id 5489  df-eprel 5495  df-fr 5544  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-bnj17 32666
This theorem is referenced by:  bnj910  32928
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