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Theorem bnj543 32065
Description: Technical lemma for bnj852 32093. 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 31877 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚))
2 bnj268 31879 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
31, 2bitri 276 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
4 bnj253 31874 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
5 bnj256 31876 . . . . . 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 302 . . . . 5 ((((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
7 bnj256 31876 . . . . . 6 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
873anbi1i 1149 . . . . 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 31868 . . . . . . 7 ((𝑓 Fn 𝑚𝜑′𝜓′) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
119, 10bitri 276 . . . . . 6 (𝜏 ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
12 bnj543.5 . . . . . . 7 (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
13 3anan32 1089 . . . . . . 7 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚) ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1412, 13bitri 276 . . . . . 6 (𝜎 ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1511, 14anbi12i 626 . . . . 5 ((𝜏𝜎) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
166, 8, 153bitr4ri 305 . . . 4 ((𝜏𝜎) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
1716anbi2i 622 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜏𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚)))
18 3anass 1087 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏𝜎)))
19 bnj252 31873 . . 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 304 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
21 df-suc 6191 . . . . . . 7 suc 𝑚 = (𝑚 ∪ {𝑚})
2221eqeq2i 2834 . . . . . 6 (𝑛 = suc 𝑚𝑛 = (𝑚 ∪ {𝑚}))
23223anbi2i 1150 . . . . 5 (((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚))
2423anbi2i 622 . . . 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 31873 . . . 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 304 . . 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 262 . . . 4 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
3127, 28, 29, 30bnj535 32062 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3226, 31sylbi 218 . 2 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3320, 32sylbi 218 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3138  cun 3933  c0 4290  {csn 4559  cop 4565   ciun 4912  suc csuc 6187   Fn wfn 6344  cfv 6349  ωcom 7568  w-bnj17 31856   predc-bnj14 31858   FrSe w-bnj15 31862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7450  ax-reg 9045
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7569  df-bnj17 31857  df-bnj14 31859  df-bnj13 31861  df-bnj15 31863
This theorem is referenced by:  bnj544  32066
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