Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj570 Structured version   Visualization version   GIF version

Theorem bnj570 34541
Description: Technical lemma for bnj852 34557. 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
bnj570.3 𝐷 = (ω ∖ {∅})
bnj570.17 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj570.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj570.21 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
bnj570.24 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj570.26 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj570.40 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
bnj570.30 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj570 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = 𝐾)
Distinct variable groups:   𝑦,𝐺   𝑦,𝑓   𝑦,𝑖
Allowed substitution hints:   𝜏(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜌(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐺(𝑓,𝑖,𝑚,𝑛,𝑝)   𝐾(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj570
StepHypRef Expression
1 bnj251 34338 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏 ∧ (𝜂𝜌))))
2 bnj570.17 . . . . . 6 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
32simp3bi 1144 . . . . 5 (𝜏𝜓′)
4 bnj570.21 . . . . . . . 8 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
54simp1bi 1142 . . . . . . 7 (𝜌𝑖 ∈ ω)
65adantl 480 . . . . . 6 ((𝜂𝜌) → 𝑖 ∈ ω)
7 bnj570.19 . . . . . . 7 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
87, 4bnj563 34379 . . . . . 6 ((𝜂𝜌) → suc 𝑖𝑚)
96, 8jca 510 . . . . 5 ((𝜂𝜌) → (𝑖 ∈ ω ∧ suc 𝑖𝑚))
10 bnj570.30 . . . . . . . 8 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
1110bnj946 34410 . . . . . . 7 (𝜓′ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
12 sp 2171 . . . . . . 7 (∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
1311, 12sylbi 216 . . . . . 6 (𝜓′ → (𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
1413imp32 417 . . . . 5 ((𝜓′ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑚)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
153, 9, 14syl2an 594 . . . 4 ((𝜏 ∧ (𝜂𝜌)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
161, 15simplbiim 503 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
17 bnj570.40 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
1817fnfund 6658 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂) → Fun 𝐺)
1918bnj721 34393 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → Fun 𝐺)
20 bnj570.26 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
2120bnj931 34406 . . . . 5 𝑓𝐺
2221a1i 11 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑓𝐺)
23 bnj667 34388 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝜏𝜂𝜌))
242bnj564 34380 . . . . . . 7 (𝜏 → dom 𝑓 = 𝑚)
25 eleq2 2817 . . . . . . . 8 (dom 𝑓 = 𝑚 → (suc 𝑖 ∈ dom 𝑓 ↔ suc 𝑖𝑚))
2625biimpar 476 . . . . . . 7 ((dom 𝑓 = 𝑚 ∧ suc 𝑖𝑚) → suc 𝑖 ∈ dom 𝑓)
2724, 8, 26syl2an 594 . . . . . 6 ((𝜏 ∧ (𝜂𝜌)) → suc 𝑖 ∈ dom 𝑓)
28273impb 1112 . . . . 5 ((𝜏𝜂𝜌) → suc 𝑖 ∈ dom 𝑓)
2923, 28syl 17 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → suc 𝑖 ∈ dom 𝑓)
3019, 22, 29bnj1502 34484 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
312simp1bi 1142 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
32 bnj252 34339 . . . . . . . . . . . . . 14 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚𝐷 ∧ (𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
3332simplbi 496 . . . . . . . . . . . . 13 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑚𝐷)
347, 33sylbi 216 . . . . . . . . . . . 12 (𝜂𝑚𝐷)
35 eldifi 4125 . . . . . . . . . . . . 13 (𝑚 ∈ (ω ∖ {∅}) → 𝑚 ∈ ω)
36 bnj570.3 . . . . . . . . . . . . 13 𝐷 = (ω ∖ {∅})
3735, 36eleq2s 2846 . . . . . . . . . . . 12 (𝑚𝐷𝑚 ∈ ω)
38 nnord 7882 . . . . . . . . . . . 12 (𝑚 ∈ ω → Ord 𝑚)
3934, 37, 383syl 18 . . . . . . . . . . 11 (𝜂 → Ord 𝑚)
4039adantr 479 . . . . . . . . . 10 ((𝜂𝜌) → Ord 𝑚)
4140, 8jca 510 . . . . . . . . 9 ((𝜂𝜌) → (Ord 𝑚 ∧ suc 𝑖𝑚))
4231, 41anim12i 611 . . . . . . . 8 ((𝜏 ∧ (𝜂𝜌)) → (𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖𝑚)))
43 fndm 6660 . . . . . . . . 9 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
44 elelsuc 6445 . . . . . . . . . 10 (suc 𝑖𝑚 → suc 𝑖 ∈ suc 𝑚)
45 ordsucelsuc 7829 . . . . . . . . . . 11 (Ord 𝑚 → (𝑖𝑚 ↔ suc 𝑖 ∈ suc 𝑚))
4645biimpar 476 . . . . . . . . . 10 ((Ord 𝑚 ∧ suc 𝑖 ∈ suc 𝑚) → 𝑖𝑚)
4744, 46sylan2 591 . . . . . . . . 9 ((Ord 𝑚 ∧ suc 𝑖𝑚) → 𝑖𝑚)
4843, 47anim12i 611 . . . . . . . 8 ((𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖𝑚)) → (dom 𝑓 = 𝑚𝑖𝑚))
49 eleq2 2817 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓𝑖𝑚))
5049biimpar 476 . . . . . . . 8 ((dom 𝑓 = 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
5142, 48, 503syl 18 . . . . . . 7 ((𝜏 ∧ (𝜂𝜌)) → 𝑖 ∈ dom 𝑓)
52513impb 1112 . . . . . 6 ((𝜏𝜂𝜌) → 𝑖 ∈ dom 𝑓)
5323, 52syl 17 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑖 ∈ dom 𝑓)
5419, 22, 53bnj1502 34484 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺𝑖) = (𝑓𝑖))
5554iuneq1d 5025 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
5616, 30, 553eqtr4d 2777 . 2 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
57 bnj570.24 . 2 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
5856, 57eqtr4di 2785 1 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wal 1531   = wceq 1533  wcel 2098  wne 2936  wral 3057  cdif 3944  cun 3945  wss 3947  c0 4324  {csn 4630  cop 4636   ciun 4998  dom cdm 5680  Ord word 6371  suc csuc 6374  Fun wfun 6545   Fn wfn 6546  cfv 6551  ωcom 7874  w-bnj17 34322   predc-bnj14 34324   FrSe w-bnj15 34328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-res 5692  df-ord 6375  df-on 6376  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-fv 6559  df-om 7875  df-bnj17 34323
This theorem is referenced by:  bnj571  34542
  Copyright terms: Public domain W3C validator