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Theorem bnj570 30949
Description: Technical lemma for bnj852 30965. 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 30742 . . . 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 30787 . . . . . 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 30819 . . . . . . 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 30814 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂) → Fun 𝐺)
1918bnj721 30801 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → Fun 𝐺)
20 bnj570.26 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
2120bnj931 30815 . . . . 5 𝑓𝐺
2221a1i 11 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑓𝐺)
23 bnj667 30796 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝜏𝜂𝜌))
242bnj564 30788 . . . . . . 7 (𝜏 → dom 𝑓 = 𝑚)
25 eleq2 2688 . . . . . . . 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 1258 . . . . 5 ((𝜏𝜂𝜌) → suc 𝑖 ∈ dom 𝑓)
2923, 28syl 17 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → suc 𝑖 ∈ dom 𝑓)
3019, 22, 29bnj1502 30892 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
312simp1bi 1074 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
32 bnj252 30743 . . . . . . . . . . . . . 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 3724 . . . . . . . . . . . . 13 (𝑚 ∈ (ω ∖ {∅}) → 𝑚 ∈ ω)
36 bnj570.3 . . . . . . . . . . . . 13 𝐷 = (ω ∖ {∅})
3735, 36eleq2s 2717 . . . . . . . . . . . 12 (𝑚𝐷𝑚 ∈ ω)
38 nnord 7058 . . . . . . . . . . . 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 5978 . . . . . . . . 9 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
44 elelsuc 5785 . . . . . . . . . 10 (suc 𝑖𝑚 → suc 𝑖 ∈ suc 𝑚)
45 ordsucelsuc 7007 . . . . . . . . . . 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 2688 . . . . . . . . 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 1258 . . . . . 6 ((𝜏𝜂𝜌) → 𝑖 ∈ dom 𝑓)
5323, 52syl 17 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑖 ∈ dom 𝑓)
5419, 22, 53bnj1502 30892 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺𝑖) = (𝑓𝑖))
5554iuneq1d 4536 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
5616, 30, 553eqtr4d 2664 . 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 2672 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 1479   = wceq 1481  wcel 1988  wne 2791  wral 2909  cdif 3564  cun 3565  wss 3567  c0 3907  {csn 4168  cop 4174   ciun 4511  dom cdm 5104  Ord word 5710  suc csuc 5713  Fun wfun 5870   Fn wfn 5871  cfv 5876  ωcom 7050  w-bnj17 30726   predc-bnj14 30728   FrSe w-bnj15 30732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-res 5116  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-om 7051  df-bnj17 30727
This theorem is referenced by:  bnj571  30950
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