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Theorem bnj535 31340
Description: Technical lemma for bnj852 31371. 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
bnj535.1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj535.2 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj535.3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj535.4 (𝜏 ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
Assertion
Ref Expression
bnj535 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′
Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑚,𝑛)   𝑅(𝑥,𝑓,𝑚,𝑛)   𝐺(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 bnj535
StepHypRef Expression
1 bnj422 31164 . . 3 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚𝑅 FrSe 𝐴𝜏))
2 bnj251 31151 . . 3 ((𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚𝑅 FrSe 𝐴𝜏) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏))))
31, 2bitri 266 . 2 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏))))
4 fvex 6388 . . . . . . . . 9 (𝑓𝑝) ∈ V
5 bnj535.1 . . . . . . . . . 10 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
6 bnj535.2 . . . . . . . . . 10 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
7 bnj535.4 . . . . . . . . . 10 (𝜏 ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
85, 6, 7bnj518 31336 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
9 iunexg 7341 . . . . . . . . 9 (((𝑓𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
104, 8, 9sylancr 581 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
11 vex 3353 . . . . . . . . 9 𝑚 ∈ V
1211bnj519 31185 . . . . . . . 8 ( 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → Fun {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
1310, 12syl 17 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏) → Fun {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
14 dmsnopg 5790 . . . . . . . 8 ( 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → dom {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {𝑚})
1510, 14syl 17 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏) → dom {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {𝑚})
1613, 15bnj1422 31288 . . . . . 6 ((𝑅 FrSe 𝐴𝜏) → {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} Fn {𝑚})
17 bnj521 31186 . . . . . . 7 (𝑚 ∩ {𝑚}) = ∅
18 fnun 6175 . . . . . . 7 (((𝑓 Fn 𝑚 ∧ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} Fn {𝑚}) ∧ (𝑚 ∩ {𝑚}) = ∅) → (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
1917, 18mpan2 682 . . . . . 6 ((𝑓 Fn 𝑚 ∧ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} Fn {𝑚}) → (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
2016, 19sylan2 586 . . . . 5 ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏)) → (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
21 bnj535.3 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
2221fneq1i 6163 . . . . 5 (𝐺 Fn (𝑚 ∪ {𝑚}) ↔ (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
2320, 22sylibr 225 . . . 4 ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏)) → 𝐺 Fn (𝑚 ∪ {𝑚}))
24 fneq2 6158 . . . 4 (𝑛 = (𝑚 ∪ {𝑚}) → (𝐺 Fn 𝑛𝐺 Fn (𝑚 ∪ {𝑚})))
2523, 24syl5ibr 237 . . 3 (𝑛 = (𝑚 ∪ {𝑚}) → ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏)) → 𝐺 Fn 𝑛))
2625imp 395 . 2 ((𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏))) → 𝐺 Fn 𝑛)
273, 26sylbi 208 1 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cun 3730  cin 3731  c0 4079  {csn 4334  cop 4340   ciun 4676  dom cdm 5277  suc csuc 5910  Fun wfun 6062   Fn wfn 6063  cfv 6068  ωcom 7263  w-bnj17 31135   predc-bnj14 31137   FrSe w-bnj15 31141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147  ax-reg 8704
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-bnj17 31136  df-bnj14 31138  df-bnj13 31140  df-bnj15 31142
This theorem is referenced by:  bnj543  31343
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