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Theorem bnj966 32495
Description: Technical lemma for bnj69 32561. 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 32320 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → Fun 𝐺)
323adant3 1133 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → Fun 𝐺)
4 opex 5322 . . . . . . 7 𝑛, 𝐶⟩ ∈ V
54snid 4552 . . . . . 6 𝑛, 𝐶⟩ ∈ {⟨𝑛, 𝐶⟩}
6 elun2 4067 . . . . . 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 2832 . . . 4 𝑛, 𝐶⟩ ∈ 𝐺
10 funopfv 6721 . . . 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 1208 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑛 = suc 𝑚)
13 simp33 1212 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑛 = suc 𝑖)
14 bnj551 32292 . . . . 5 ((𝑛 = suc 𝑚𝑛 = suc 𝑖) → 𝑚 = 𝑖)
1512, 13, 14syl2anc 587 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑚 = 𝑖)
16 suceq 6237 . . . . . . . 8 (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖)
1716eqeq2d 2749 . . . . . . 7 (𝑚 = 𝑖 → (𝑛 = suc 𝑚𝑛 = suc 𝑖))
1817biimpac 482 . . . . . 6 ((𝑛 = suc 𝑚𝑚 = 𝑖) → 𝑛 = suc 𝑖)
1918fveq2d 6678 . . . . 5 ((𝑛 = suc 𝑚𝑚 = 𝑖) → (𝐺𝑛) = (𝐺‘suc 𝑖))
20 bnj966.12 . . . . . . 7 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
21 fveq2 6674 . . . . . . . 8 (𝑚 = 𝑖 → (𝑓𝑚) = (𝑓𝑖))
2221bnj1113 32336 . . . . . . 7 (𝑚 = 𝑖 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2320, 22syl5eq 2785 . . . . . 6 (𝑚 = 𝑖𝐶 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2423adantl 485 . . . . 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 587 . . 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 235 . 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 1133 . . . . 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 32355 . . . . . . 7 (𝜒𝑓 Fn 𝑛)
32313ad2ant1 1134 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
33323ad2ant2 1135 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑓 Fn 𝑛)
34 simp23 1209 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → 𝑝 = suc 𝑛)
3529, 33, 34, 13bnj951 32326 . . . 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 32318 . . . . . . . 8 (𝑛𝐷𝑛 ∈ ω)
3830, 37bnj769 32312 . . . . . . 7 (𝜒𝑛 ∈ ω)
39383ad2ant1 1134 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑛 ∈ ω)
40 simp3 1139 . . . . . 6 ((𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖) → 𝑛 = suc 𝑖)
4139, 40bnj240 32248 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝑛 ∈ ω ∧ 𝑛 = suc 𝑖))
42 vex 3402 . . . . . . 7 𝑖 ∈ V
4342bnj216 32281 . . . . . 6 (𝑛 = suc 𝑖𝑖𝑛)
4443adantl 485 . . . . 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 32301 . . . . . . 7 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛))
4746anim1i 618 . . . . . 6 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) ∧ 𝑖𝑛) → ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
48 df-bnj17 32236 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝑖𝑛))
4947, 48sylibr 237 . . . . 5 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑛 = suc 𝑖) ∧ 𝑖𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝑖𝑛))
508bnj945 32324 . . . . 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 587 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺𝑖) = (𝑓𝑖))
5320, 8bnj958 32491 . . . . 5 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
5453bnj956 32327 . . . 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 260 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3398  cdif 3840  cun 3841  c0 4211  {csn 4516  cop 4522   ciun 4881  suc csuc 6174  Fun wfun 6333   Fn wfn 6334  cfv 6339  ωcom 7599  w-bnj17 32235   predc-bnj14 32237   FrSe w-bnj15 32241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479  ax-reg 9129
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-id 5429  df-eprel 5434  df-fr 5483  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-res 5537  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-fv 6347  df-bnj17 32236
This theorem is referenced by:  bnj910  32499
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