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Theorem bnj570 31789
Description: Technical lemma for bnj852 31805. 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 31585 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏 ∧ (𝜂𝜌))))
2 bnj570.17 . . . . . 6 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
32simp3bi 1140 . . . . 5 (𝜏𝜓′)
4 bnj570.21 . . . . . . . 8 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
54simp1bi 1138 . . . . . . 7 (𝜌𝑖 ∈ ω)
65adantl 482 . . . . . 6 ((𝜂𝜌) → 𝑖 ∈ ω)
7 bnj570.19 . . . . . . 7 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
87, 4bnj563 31627 . . . . . 6 ((𝜂𝜌) → suc 𝑖𝑚)
96, 8jca 512 . . . . 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 31659 . . . . . . 7 (𝜓′ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
12 sp 2148 . . . . . . 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 218 . . . . . 6 (𝜓′ → (𝑖 ∈ ω → (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
1413imp32 419 . . . . 5 ((𝜓′ ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑚)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
153, 9, 14syl2an 595 . . . 4 ((𝜏 ∧ (𝜂𝜌)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
161, 15simplbiim 505 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
17 bnj570.40 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
1817bnj930 31654 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂) → Fun 𝐺)
1918bnj721 31641 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → Fun 𝐺)
20 bnj570.26 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
2120bnj931 31655 . . . . 5 𝑓𝐺
2221a1i 11 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑓𝐺)
23 bnj667 31636 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝜏𝜂𝜌))
242bnj564 31628 . . . . . . 7 (𝜏 → dom 𝑓 = 𝑚)
25 eleq2 2873 . . . . . . . 8 (dom 𝑓 = 𝑚 → (suc 𝑖 ∈ dom 𝑓 ↔ suc 𝑖𝑚))
2625biimpar 478 . . . . . . 7 ((dom 𝑓 = 𝑚 ∧ suc 𝑖𝑚) → suc 𝑖 ∈ dom 𝑓)
2724, 8, 26syl2an 595 . . . . . 6 ((𝜏 ∧ (𝜂𝜌)) → suc 𝑖 ∈ dom 𝑓)
28273impb 1108 . . . . 5 ((𝜏𝜂𝜌) → suc 𝑖 ∈ dom 𝑓)
2923, 28syl 17 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → suc 𝑖 ∈ dom 𝑓)
3019, 22, 29bnj1502 31732 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
312simp1bi 1138 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
32 bnj252 31586 . . . . . . . . . . . . . 14 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚𝐷 ∧ (𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
3332simplbi 498 . . . . . . . . . . . . 13 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑚𝐷)
347, 33sylbi 218 . . . . . . . . . . . 12 (𝜂𝑚𝐷)
35 eldifi 4030 . . . . . . . . . . . . 13 (𝑚 ∈ (ω ∖ {∅}) → 𝑚 ∈ ω)
36 bnj570.3 . . . . . . . . . . . . 13 𝐷 = (ω ∖ {∅})
3735, 36eleq2s 2903 . . . . . . . . . . . 12 (𝑚𝐷𝑚 ∈ ω)
38 nnord 7451 . . . . . . . . . . . 12 (𝑚 ∈ ω → Ord 𝑚)
3934, 37, 383syl 18 . . . . . . . . . . 11 (𝜂 → Ord 𝑚)
4039adantr 481 . . . . . . . . . 10 ((𝜂𝜌) → Ord 𝑚)
4140, 8jca 512 . . . . . . . . 9 ((𝜂𝜌) → (Ord 𝑚 ∧ suc 𝑖𝑚))
4231, 41anim12i 612 . . . . . . . 8 ((𝜏 ∧ (𝜂𝜌)) → (𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖𝑚)))
43 fndm 6332 . . . . . . . . 9 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
44 elelsuc 6145 . . . . . . . . . 10 (suc 𝑖𝑚 → suc 𝑖 ∈ suc 𝑚)
45 ordsucelsuc 7400 . . . . . . . . . . 11 (Ord 𝑚 → (𝑖𝑚 ↔ suc 𝑖 ∈ suc 𝑚))
4645biimpar 478 . . . . . . . . . 10 ((Ord 𝑚 ∧ suc 𝑖 ∈ suc 𝑚) → 𝑖𝑚)
4744, 46sylan2 592 . . . . . . . . 9 ((Ord 𝑚 ∧ suc 𝑖𝑚) → 𝑖𝑚)
4843, 47anim12i 612 . . . . . . . 8 ((𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖𝑚)) → (dom 𝑓 = 𝑚𝑖𝑚))
49 eleq2 2873 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓𝑖𝑚))
5049biimpar 478 . . . . . . . 8 ((dom 𝑓 = 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
5142, 48, 503syl 18 . . . . . . 7 ((𝜏 ∧ (𝜂𝜌)) → 𝑖 ∈ dom 𝑓)
52513impb 1108 . . . . . 6 ((𝜏𝜂𝜌) → 𝑖 ∈ dom 𝑓)
5323, 52syl 17 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → 𝑖 ∈ dom 𝑓)
5419, 22, 53bnj1502 31732 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺𝑖) = (𝑓𝑖))
5554iuneq1d 4857 . . 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, 57syl6eqr 2851 1 ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080  wal 1523   = wceq 1525  wcel 2083  wne 2986  wral 3107  cdif 3862  cun 3863  wss 3865  c0 4217  {csn 4478  cop 4484   ciun 4831  dom cdm 5450  Ord word 6072  suc csuc 6075  Fun wfun 6226   Fn wfn 6227  cfv 6232  ωcom 7443  w-bnj17 31569   predc-bnj14 31571   FrSe w-bnj15 31575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-res 5462  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-fv 6240  df-om 7444  df-bnj17 31570
This theorem is referenced by:  bnj571  31790
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