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Theorem bnj545 34202
Description: Technical lemma for bnj852 34228. 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 1143 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
3 bnj545.5 . . . . . . . . . 10 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
43simp1bi 1143 . . . . . . . . 9 (𝜎𝑚𝐷)
52, 4anim12i 611 . . . . . . . 8 ((𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
653adant1 1128 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
7 bnj545.2 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
87bnj529 34048 . . . . . . . 8 (𝑚𝐷 → ∅ ∈ 𝑚)
9 fndm 6653 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
10 eleq2 2820 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑚))
1110biimparc 478 . . . . . . . 8 ((∅ ∈ 𝑚 ∧ dom 𝑓 = 𝑚) → ∅ ∈ dom 𝑓)
128, 9, 11syl2anr 595 . . . . . . 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 6651 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
1613, 15jca 510 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜎) → (∅ ∈ dom 𝑓 ∧ Fun 𝐺))
17 bnj545.3 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
1817bnj931 34077 . . . . 5 𝑓𝐺
1916, 18jctil 518 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
20 df-3an 1087 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺))
21 3anrot 1098 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
22 ancom 459 . . . . 5 (((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2320, 21, 223bitr3i 300 . . . 4 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2419, 23sylibr 233 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
25 funssfv 6913 . . 3 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) → (𝐺‘∅) = (𝑓‘∅))
2624, 25syl 17 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝐺‘∅) = (𝑓‘∅))
271simp2bi 1144 . . 3 (𝜏𝜑′)
28273ad2ant2 1132 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑′)
29 bnj545.1 . . . 4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
30 eqtr 2753 . . . 4 (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3129, 30sylan2b 592 . . 3 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
32 bnj545.7 . . 3 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3331, 32sylibr 233 . 2 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → 𝜑″)
3426, 28, 33syl2anc 582 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  cdif 3946  cun 3947  wss 3949  c0 4323  {csn 4629  cop 4635   ciun 4998  dom cdm 5677  suc csuc 6367  Fun wfun 6538   Fn wfn 6539  cfv 6544  ωcom 7859   predc-bnj14 33995   FrSe w-bnj15 33999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-om 7860
This theorem is referenced by:  bnj600  34226  bnj908  34238
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