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Theorem bnj543 34907
Description: Technical lemma for bnj852 34935. 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
bnj543.1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj543.2 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj543.3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj543.4 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj543.5 (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
Assertion
Ref Expression
bnj543 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′
Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑚,𝑛)   𝑅(𝑥,𝑓,𝑚,𝑛)   𝐺(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj543
StepHypRef Expression
1 bnj257 34721 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚))
2 bnj268 34723 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
31, 2bitri 275 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
4 bnj253 34718 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
5 bnj256 34720 . . . . . 6 (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
63, 4, 53bitr3i 301 . . . . 5 ((((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
7 bnj256 34720 . . . . . 6 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
873anbi1i 1158 . . . . 5 (((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
9 bnj543.4 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
10 bnj170 34712 . . . . . . 7 ((𝑓 Fn 𝑚𝜑′𝜓′) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
119, 10bitri 275 . . . . . 6 (𝜏 ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
12 bnj543.5 . . . . . . 7 (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
13 3anan32 1097 . . . . . . 7 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚) ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1412, 13bitri 275 . . . . . 6 (𝜎 ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1511, 14anbi12i 628 . . . . 5 ((𝜏𝜎) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
166, 8, 153bitr4ri 304 . . . 4 ((𝜏𝜎) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
1716anbi2i 623 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜏𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚)))
18 3anass 1095 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏𝜎)))
19 bnj252 34717 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚)))
2017, 18, 193bitr4i 303 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
21 df-suc 6390 . . . . . . 7 suc 𝑚 = (𝑚 ∪ {𝑚})
2221eqeq2i 2750 . . . . . 6 (𝑛 = suc 𝑚𝑛 = (𝑚 ∪ {𝑚}))
23223anbi2i 1159 . . . . 5 (((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚))
2423anbi2i 623 . . . 4 ((𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚)))
25 bnj252 34717 . . . 4 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚)))
2624, 19, 253bitr4i 303 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚))
27 bnj543.1 . . . 4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
28 bnj543.2 . . . 4 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
29 bnj543.3 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
30 biid 261 . . . 4 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
3127, 28, 29, 30bnj535 34904 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3226, 31sylbi 217 . 2 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3320, 32sylbi 217 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cun 3949  c0 4333  {csn 4626  cop 4632   ciun 4991  suc csuc 6386   Fn wfn 6556  cfv 6561  ωcom 7887  w-bnj17 34700   predc-bnj14 34702   FrSe w-bnj15 34706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-om 7888  df-bnj17 34701  df-bnj14 34703  df-bnj13 34705  df-bnj15 34707
This theorem is referenced by:  bnj544  34908
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