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Theorem bnj543 31411
Description: Technical lemma for bnj852 31439. 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 31224 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚))
2 bnj268 31226 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
31, 2bitri 266 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
4 bnj253 31221 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
5 bnj256 31223 . . . . . 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 292 . . . . 5 ((((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
7 bnj256 31223 . . . . . 6 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
873anbi1i 1196 . . . . 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 31215 . . . . . . 7 ((𝑓 Fn 𝑚𝜑′𝜓′) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
119, 10bitri 266 . . . . . 6 (𝜏 ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
12 bnj543.5 . . . . . . 7 (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
13 3anan32 1118 . . . . . . 7 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚) ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1412, 13bitri 266 . . . . . 6 (𝜎 ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1511, 14anbi12i 620 . . . . 5 ((𝜏𝜎) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
166, 8, 153bitr4ri 295 . . . 4 ((𝜏𝜎) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
1716anbi2i 616 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜏𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚)))
18 3anass 1116 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏𝜎)))
19 bnj252 31220 . . 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 294 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
21 df-suc 5914 . . . . . . 7 suc 𝑚 = (𝑚 ∪ {𝑚})
2221eqeq2i 2777 . . . . . 6 (𝑛 = suc 𝑚𝑛 = (𝑚 ∪ {𝑚}))
23223anbi2i 1197 . . . . 5 (((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚))
2423anbi2i 616 . . . 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 31220 . . . 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 294 . . 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 252 . . . 4 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
3127, 28, 29, 30bnj535 31408 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3226, 31sylbi 208 . 2 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3320, 32sylbi 208 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  cun 3730  c0 4079  {csn 4334  cop 4340   ciun 4676  suc csuc 5910   Fn wfn 6063  cfv 6068  ωcom 7263  w-bnj17 31203   predc-bnj14 31205   FrSe w-bnj15 31209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147  ax-reg 8704
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-bnj17 31204  df-bnj14 31206  df-bnj13 31208  df-bnj15 31210
This theorem is referenced by:  bnj544  31412
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