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Theorem bnj545 34888
Description: Technical lemma for bnj852 34914. 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
bnj545.1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj545.2 𝐷 = (ω ∖ {∅})
bnj545.3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj545.4 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj545.5 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj545.6 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
bnj545.7 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
Assertion
Ref Expression
bnj545 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)

Proof of Theorem bnj545
StepHypRef Expression
1 bnj545.4 . . . . . . . . . 10 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
21simp1bi 1144 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
3 bnj545.5 . . . . . . . . . 10 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
43simp1bi 1144 . . . . . . . . 9 (𝜎𝑚𝐷)
52, 4anim12i 613 . . . . . . . 8 ((𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
653adant1 1129 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
7 bnj545.2 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
87bnj529 34734 . . . . . . . 8 (𝑚𝐷 → ∅ ∈ 𝑚)
9 fndm 6672 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
10 eleq2 2828 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑚))
1110biimparc 479 . . . . . . . 8 ((∅ ∈ 𝑚 ∧ dom 𝑓 = 𝑚) → ∅ ∈ dom 𝑓)
128, 9, 11syl2anr 597 . . . . . . 7 ((𝑓 Fn 𝑚𝑚𝐷) → ∅ ∈ dom 𝑓)
136, 12syl 17 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → ∅ ∈ dom 𝑓)
14 bnj545.6 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
1514fnfund 6670 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
1613, 15jca 511 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜎) → (∅ ∈ dom 𝑓 ∧ Fun 𝐺))
17 bnj545.3 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
1817bnj931 34763 . . . . 5 𝑓𝐺
1916, 18jctil 519 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
20 df-3an 1088 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺))
21 3anrot 1099 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
22 ancom 460 . . . . 5 (((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2320, 21, 223bitr3i 301 . . . 4 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2419, 23sylibr 234 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
25 funssfv 6928 . . 3 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) → (𝐺‘∅) = (𝑓‘∅))
2624, 25syl 17 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝐺‘∅) = (𝑓‘∅))
271simp2bi 1145 . . 3 (𝜏𝜑′)
28273ad2ant2 1133 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑′)
29 bnj545.1 . . . 4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
30 eqtr 2758 . . . 4 (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3129, 30sylan2b 594 . . 3 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
32 bnj545.7 . . 3 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3331, 32sylibr 234 . 2 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → 𝜑″)
3426, 28, 33syl2anc 584 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cdif 3960  cun 3961  wss 3963  c0 4339  {csn 4631  cop 4637   ciun 4996  dom cdm 5689  suc csuc 6388  Fun wfun 6557   Fn wfn 6558  cfv 6563  ωcom 7887   predc-bnj14 34681   FrSe w-bnj15 34685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-om 7888
This theorem is referenced by:  bnj600  34912  bnj908  34924
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