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Theorem bnj570 32287
Description: Technical lemma for bnj852 32303. 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 32082 . . . 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 485 . . . . . 6 ((𝜂𝜌) → 𝑖 ∈ ω)
7 bnj570.19 . . . . . . 7 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
87, 4bnj563 32124 . . . . . 6 ((𝜂𝜌) → suc 𝑖𝑚)
96, 8jca 515 . . . . 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 32156 . . . . . . 7 (𝜓′ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
12 sp 2180 . . . . . . 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 220 . . . . . 6 (𝜓′ → (𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
1413imp32 422 . . . . 5 ((𝜓′ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑚)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
153, 9, 14syl2an 598 . . . 4 ((𝜏 ∧ (𝜂𝜌)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
161, 15simplbiim 508 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
17 bnj570.40 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
1817bnj930 32151 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂) → Fun 𝐺)
1918bnj721 32138 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → Fun 𝐺)
20 bnj570.26 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
2120bnj931 32152 . . . . 5 𝑓𝐺
2221a1i 11 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑓𝐺)
23 bnj667 32133 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝜏𝜂𝜌))
242bnj564 32125 . . . . . . 7 (𝜏 → dom 𝑓 = 𝑚)
25 eleq2 2878 . . . . . . . 8 (dom 𝑓 = 𝑚 → (suc 𝑖 ∈ dom 𝑓 ↔ suc 𝑖𝑚))
2625biimpar 481 . . . . . . 7 ((dom 𝑓 = 𝑚 ∧ suc 𝑖𝑚) → suc 𝑖 ∈ dom 𝑓)
2724, 8, 26syl2an 598 . . . . . 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 32230 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
312simp1bi 1142 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
32 bnj252 32083 . . . . . . . . . . . . . 14 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚𝐷 ∧ (𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
3332simplbi 501 . . . . . . . . . . . . 13 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑚𝐷)
347, 33sylbi 220 . . . . . . . . . . . 12 (𝜂𝑚𝐷)
35 eldifi 4054 . . . . . . . . . . . . 13 (𝑚 ∈ (ω ∖ {∅}) → 𝑚 ∈ ω)
36 bnj570.3 . . . . . . . . . . . . 13 𝐷 = (ω ∖ {∅})
3735, 36eleq2s 2908 . . . . . . . . . . . 12 (𝑚𝐷𝑚 ∈ ω)
38 nnord 7568 . . . . . . . . . . . 12 (𝑚 ∈ ω → Ord 𝑚)
3934, 37, 383syl 18 . . . . . . . . . . 11 (𝜂 → Ord 𝑚)
4039adantr 484 . . . . . . . . . 10 ((𝜂𝜌) → Ord 𝑚)
4140, 8jca 515 . . . . . . . . 9 ((𝜂𝜌) → (Ord 𝑚 ∧ suc 𝑖𝑚))
4231, 41anim12i 615 . . . . . . . 8 ((𝜏 ∧ (𝜂𝜌)) → (𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖𝑚)))
43 fndm 6425 . . . . . . . . 9 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
44 elelsuc 6231 . . . . . . . . . 10 (suc 𝑖𝑚 → suc 𝑖 ∈ suc 𝑚)
45 ordsucelsuc 7517 . . . . . . . . . . 11 (Ord 𝑚 → (𝑖𝑚 ↔ suc 𝑖 ∈ suc 𝑚))
4645biimpar 481 . . . . . . . . . 10 ((Ord 𝑚 ∧ suc 𝑖 ∈ suc 𝑚) → 𝑖𝑚)
4744, 46sylan2 595 . . . . . . . . 9 ((Ord 𝑚 ∧ suc 𝑖𝑚) → 𝑖𝑚)
4843, 47anim12i 615 . . . . . . . 8 ((𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖𝑚)) → (dom 𝑓 = 𝑚𝑖𝑚))
49 eleq2 2878 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓𝑖𝑚))
5049biimpar 481 . . . . . . . 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 32230 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺𝑖) = (𝑓𝑖))
5554iuneq1d 4908 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
5616, 30, 553eqtr4d 2843 . 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 2851 1 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2111  wne 2987  wral 3106  cdif 3878  cun 3879  wss 3881  c0 4243  {csn 4525  cop 4531   ciun 4881  dom cdm 5519  Ord word 6158  suc csuc 6161  Fun wfun 6318   Fn wfn 6319  cfv 6324  ωcom 7560  w-bnj17 32066   predc-bnj14 32068   FrSe w-bnj15 32072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-om 7561  df-bnj17 32067
This theorem is referenced by:  bnj571  32288
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