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Theorem bnj535 31289
Description: Technical lemma for bnj852 31320. 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 31112 . . 3 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚𝑅 FrSe 𝐴𝜏))
2 bnj251 31099 . . 3 ((𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚𝑅 FrSe 𝐴𝜏) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏))))
31, 2bitri 264 . 2 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏))))
4 fvex 6364 . . . . . . . . 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 31285 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
9 iunexg 7310 . . . . . . . . 9 (((𝑓𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
104, 8, 9sylancr 698 . . . . . . . 8 ((𝑅 FrSe 𝐴𝜏) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
11 vex 3344 . . . . . . . . 9 𝑚 ∈ V
1211bnj519 31133 . . . . . . . 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 5766 . . . . . . . 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 31237 . . . . . 6 ((𝑅 FrSe 𝐴𝜏) → {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} Fn {𝑚})
17 bnj521 31134 . . . . . . 7 (𝑚 ∩ {𝑚}) = ∅
18 fnun 6159 . . . . . . 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 709 . . . . . 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 492 . . . . 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 6147 . . . . 5 (𝐺 Fn (𝑚 ∪ {𝑚}) ↔ (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
2320, 22sylibr 224 . . . 4 ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏)) → 𝐺 Fn (𝑚 ∪ {𝑚}))
24 fneq2 6142 . . . 4 (𝑛 = (𝑚 ∪ {𝑚}) → (𝐺 Fn 𝑛𝐺 Fn (𝑚 ∪ {𝑚})))
2523, 24syl5ibr 236 . . 3 (𝑛 = (𝑚 ∪ {𝑚}) → ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏)) → 𝐺 Fn 𝑛))
2625imp 444 . 2 ((𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴𝜏))) → 𝐺 Fn 𝑛)
273, 26sylbi 207 1 ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2140  wral 3051  Vcvv 3341  cun 3714  cin 3715  c0 4059  {csn 4322  cop 4328   ciun 4673  dom cdm 5267  suc csuc 5887  Fun wfun 6044   Fn wfn 6045  cfv 6050  ωcom 7232  w-bnj17 31083   predc-bnj14 31085   FrSe w-bnj15 31089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116  ax-reg 8665
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-om 7233  df-bnj17 31084  df-bnj14 31086  df-bnj13 31088  df-bnj15 31090
This theorem is referenced by:  bnj543  31292
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