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Theorem bnj966 30268
Description: Technical lemma for bnj69 30332. 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)
21bnj930 30094 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → Fun 𝐺)
323adant3 1074 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → Fun 𝐺)
4 opex 4859 . . . . . . 7 𝑛, 𝐶⟩ ∈ V
54snid 4155 . . . . . 6 𝑛, 𝐶⟩ ∈ {⟨𝑛, 𝐶⟩}
6 elun2 3743 . . . . . 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 2687 . . . 4 𝑛, 𝐶⟩ ∈ 𝐺
10 funopfv 6145 . . . 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 1088 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑛 = suc 𝑚)
13 simp33 1092 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑛 = suc 𝑖)
14 bnj551 30066 . . . . 5 ((𝑛 = suc 𝑚𝑛 = suc 𝑖) → 𝑚 = 𝑖)
1512, 13, 14syl2anc 691 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑚 = 𝑖)
16 suceq 5707 . . . . . . . 8 (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖)
1716eqeq2d 2620 . . . . . . 7 (𝑚 = 𝑖 → (𝑛 = suc 𝑚𝑛 = suc 𝑖))
1817biimpac 502 . . . . . 6 ((𝑛 = suc 𝑚𝑚 = 𝑖) → 𝑛 = suc 𝑖)
1918fveq2d 6107 . . . . 5 ((𝑛 = suc 𝑚𝑚 = 𝑖) → (𝐺𝑛) = (𝐺‘suc 𝑖))
20 bnj966.12 . . . . . . 7 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
21 fveq2 6103 . . . . . . . 8 (𝑚 = 𝑖 → (𝑓𝑚) = (𝑓𝑖))
2221bnj1113 30110 . . . . . . 7 (𝑚 = 𝑖 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2320, 22syl5eq 2656 . . . . . 6 (𝑚 = 𝑖𝐶 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2423adantl 481 . . . . 5 ((𝑛 = suc 𝑚𝑚 = 𝑖) → 𝐶 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2519, 24eqeq12d 2625 . . . 4 ((𝑛 = suc 𝑚𝑚 = 𝑖) → ((𝐺𝑛) = 𝐶 ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2612, 15, 25syl2anc 691 . . 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 221 . 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 1074 . . . . 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 30129 . . . . . . 7 (𝜒𝑓 Fn 𝑛)
32313ad2ant1 1075 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
33323ad2ant2 1076 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑓 Fn 𝑛)
34 simp23 1089 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑝 = suc 𝑛)
3529, 33, 34, 13bnj951 30100 . . . 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 30092 . . . . . . . 8 (𝑛𝐷𝑛 ∈ ω)
3830, 37bnj769 30086 . . . . . . 7 (𝜒𝑛 ∈ ω)
39383ad2ant1 1075 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛 ∈ ω)
40 simp3 1056 . . . . . 6 ((𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖) → 𝑛 = suc 𝑖)
4139, 40bnj240 30018 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝑛 ∈ ω ∧ 𝑛 = suc 𝑖))
42 vex 3176 . . . . . . 7 𝑖 ∈ V
4342bnj216 30054 . . . . . 6 (𝑛 = suc 𝑖𝑖𝑛)
4443adantl 481 . . . . 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 30075 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛))
4746anim1i 590 . . . . . 6 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) ∧ 𝑖𝑛) → ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
48 df-bnj17 30006 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
4947, 48sylibr 223 . . . . 5 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) ∧ 𝑖𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛))
508bnj945 30098 . . . . 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 691 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺𝑖) = (𝑓𝑖))
5320, 8bnj958 30264 . . . . 5 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
5453bnj956 30101 . . . 4 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
5554eqeq2d 2620 . . 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 246 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  Vcvv 3173  cdif 3537  cun 3538  c0 3874  {csn 4125  cop 4131   ciun 4455  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  cfv 5804  ωcom 6957  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-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-reg 8380
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-eprel 4949  df-id 4953  df-fr 4997  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-bnj17 30006
This theorem is referenced by:  bnj910  30272
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