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Theorem bnj535 31783
Description: Technical lemma for bnj852 31814. 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 31607 . . 3 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚𝑅 FrSe 𝐴𝜏))
2 bnj251 31594 . . 3 ((𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚𝑅 FrSe 𝐴𝜏) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏))))
31, 2bitri 276 . 2 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏))))
4 fvex 6556 . . . . . . . . 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 31779 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
9 iunexg 7525 . . . . . . . . 9 (((𝑓𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
104, 8, 9sylancr 587 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
11 vex 3440 . . . . . . . . 9 𝑚 ∈ V
1211bnj519 31628 . . . . . . . 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 5950 . . . . . . . 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 31731 . . . . . 6 ((𝑅 FrSe 𝐴𝜏) → {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} Fn {𝑚})
17 bnj521 31629 . . . . . . 7 (𝑚 ∩ {𝑚}) = ∅
18 fnun 6338 . . . . . . 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 687 . . . . . 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 592 . . . . 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 6325 . . . . 5 (𝐺 Fn (𝑚 ∪ {𝑚}) ↔ (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
2320, 22sylibr 235 . . . 4 ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏)) → 𝐺 Fn (𝑚 ∪ {𝑚}))
24 fneq2 6320 . . . 4 (𝑛 = (𝑚 ∪ {𝑚}) → (𝐺 Fn 𝑛𝐺 Fn (𝑚 ∪ {𝑚})))
2523, 24syl5ibr 247 . . 3 (𝑛 = (𝑚 ∪ {𝑚}) → ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏)) → 𝐺 Fn 𝑛))
2625imp 407 . 2 ((𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏))) → 𝐺 Fn 𝑛)
273, 26sylbi 218 1 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  Vcvv 3437  cun 3861  cin 3862  c0 4215  {csn 4476  cop 4482   ciun 4829  dom cdm 5448  suc csuc 6073  Fun wfun 6224   Fn wfn 6225  cfv 6230  ωcom 7441  w-bnj17 31578   predc-bnj14 31580   FrSe w-bnj15 31584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pr 5226  ax-un 7324  ax-reg 8907
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-om 7442  df-bnj17 31579  df-bnj14 31581  df-bnj13 31583  df-bnj15 31585
This theorem is referenced by:  bnj543  31786
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