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 30710
Description: Technical lemma for bnj852 30726. 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 30502 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏 ∧ (𝜂𝜌))))
2 bnj570.17 . . . . . 6 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
32simp3bi 1076 . . . . 5 (𝜏𝜓′)
4 bnj570.21 . . . . . . . 8 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
54simp1bi 1074 . . . . . . 7 (𝜌𝑖 ∈ ω)
65adantl 482 . . . . . 6 ((𝜂𝜌) → 𝑖 ∈ ω)
7 bnj570.19 . . . . . . 7 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
87, 4bnj563 30548 . . . . . 6 ((𝜂𝜌) → suc 𝑖𝑚)
96, 8jca 554 . . . . 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 30580 . . . . . . 7 (𝜓′ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
12 sp 2051 . . . . . . 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 207 . . . . . 6 (𝜓′ → (𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
1413imp32 449 . . . . 5 ((𝜓′ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑚)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
153, 9, 14syl2an 494 . . . 4 ((𝜏 ∧ (𝜂𝜌)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
161, 15simplbiim 658 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
17 bnj570.40 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
1817bnj930 30575 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂) → Fun 𝐺)
1918bnj721 30562 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → Fun 𝐺)
20 bnj570.26 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
2120bnj931 30576 . . . . 5 𝑓𝐺
2221a1i 11 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑓𝐺)
23 bnj667 30557 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝜏𝜂𝜌))
242bnj564 30549 . . . . . . 7 (𝜏 → dom 𝑓 = 𝑚)
25 eleq2 2687 . . . . . . . 8 (dom 𝑓 = 𝑚 → (suc 𝑖 ∈ dom 𝑓 ↔ suc 𝑖𝑚))
2625biimpar 502 . . . . . . 7 ((dom 𝑓 = 𝑚 ∧ suc 𝑖𝑚) → suc 𝑖 ∈ dom 𝑓)
2724, 8, 26syl2an 494 . . . . . 6 ((𝜏 ∧ (𝜂𝜌)) → suc 𝑖 ∈ dom 𝑓)
28273impb 1257 . . . . 5 ((𝜏𝜂𝜌) → suc 𝑖 ∈ dom 𝑓)
2923, 28syl 17 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → suc 𝑖 ∈ dom 𝑓)
3019, 22, 29bnj1502 30653 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
312simp1bi 1074 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
32 bnj252 30503 . . . . . . . . . . . . . 14 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚𝐷 ∧ (𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
3332simplbi 476 . . . . . . . . . . . . 13 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑚𝐷)
347, 33sylbi 207 . . . . . . . . . . . 12 (𝜂𝑚𝐷)
35 eldifi 3715 . . . . . . . . . . . . 13 (𝑚 ∈ (ω ∖ {∅}) → 𝑚 ∈ ω)
36 bnj570.3 . . . . . . . . . . . . 13 𝐷 = (ω ∖ {∅})
3735, 36eleq2s 2716 . . . . . . . . . . . 12 (𝑚𝐷𝑚 ∈ ω)
38 nnord 7027 . . . . . . . . . . . 12 (𝑚 ∈ ω → Ord 𝑚)
3934, 37, 383syl 18 . . . . . . . . . . 11 (𝜂 → Ord 𝑚)
4039adantr 481 . . . . . . . . . 10 ((𝜂𝜌) → Ord 𝑚)
4140, 8jca 554 . . . . . . . . 9 ((𝜂𝜌) → (Ord 𝑚 ∧ suc 𝑖𝑚))
4231, 41anim12i 589 . . . . . . . 8 ((𝜏 ∧ (𝜂𝜌)) → (𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖𝑚)))
43 fndm 5953 . . . . . . . . 9 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
44 elelsuc 5761 . . . . . . . . . 10 (suc 𝑖𝑚 → suc 𝑖 ∈ suc 𝑚)
45 ordsucelsuc 6976 . . . . . . . . . . 11 (Ord 𝑚 → (𝑖𝑚 ↔ suc 𝑖 ∈ suc 𝑚))
4645biimpar 502 . . . . . . . . . 10 ((Ord 𝑚 ∧ suc 𝑖 ∈ suc 𝑚) → 𝑖𝑚)
4744, 46sylan2 491 . . . . . . . . 9 ((Ord 𝑚 ∧ suc 𝑖𝑚) → 𝑖𝑚)
4843, 47anim12i 589 . . . . . . . 8 ((𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖𝑚)) → (dom 𝑓 = 𝑚𝑖𝑚))
49 eleq2 2687 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓𝑖𝑚))
5049biimpar 502 . . . . . . . 8 ((dom 𝑓 = 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
5142, 48, 503syl 18 . . . . . . 7 ((𝜏 ∧ (𝜂𝜌)) → 𝑖 ∈ dom 𝑓)
52513impb 1257 . . . . . 6 ((𝜏𝜂𝜌) → 𝑖 ∈ dom 𝑓)
5323, 52syl 17 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑖 ∈ dom 𝑓)
5419, 22, 53bnj1502 30653 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺𝑖) = (𝑓𝑖))
5554iuneq1d 4516 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
5616, 30, 553eqtr4d 2665 . 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, 57syl6eqr 2673 1 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wne 2790  wral 2907  cdif 3556  cun 3557  wss 3559  c0 3896  {csn 4153  cop 4159   ciun 4490  dom cdm 5079  Ord word 5686  suc csuc 5689  Fun wfun 5846   Fn wfn 5847  cfv 5852  ωcom 7019  w-bnj17 30486   predc-bnj14 30488   FrSe w-bnj15 30492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-om 7020  df-bnj17 30487
This theorem is referenced by:  bnj571  30711
  Copyright terms: Public domain W3C validator