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Theorem bnj543 31089
Description: Technical lemma for bnj852 31117. 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 30901 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚))
2 bnj268 30903 . . . . . . 7 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑓 Fn 𝑚𝑛 = suc 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
31, 2bitri 264 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
4 bnj253 30898 . . . . . 6 (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
5 bnj256 30900 . . . . . 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 290 . . . . 5 ((((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
7 bnj256 30900 . . . . . 6 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
873anbi1i 1272 . . . . 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 30892 . . . . . . 7 ((𝑓 Fn 𝑚𝜑′𝜓′) ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
119, 10bitri 264 . . . . . 6 (𝜏 ↔ ((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚))
12 bnj543.5 . . . . . . 7 (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))
13 3anan32 1068 . . . . . . 7 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚) ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1412, 13bitri 264 . . . . . 6 (𝜎 ↔ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚))
1511, 14anbi12i 733 . . . . 5 ((𝜏𝜎) ↔ (((𝜑′𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚)))
166, 8, 153bitr4ri 293 . . . 4 ((𝜏𝜎) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
1716anbi2i 730 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜏𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚)))
18 3anass 1059 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏𝜎)))
19 bnj252 30897 . . 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 292 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚))
21 df-suc 5767 . . . . . . 7 suc 𝑚 = (𝑚 ∪ {𝑚})
2221eqeq2i 2663 . . . . . 6 (𝑛 = suc 𝑚𝑛 = (𝑚 ∪ {𝑚}))
23223anbi2i 1273 . . . . 5 (((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) ↔ ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚))
2423anbi2i 730 . . . 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 30897 . . . 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 292 . . 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 251 . . . 4 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
3127, 28, 29, 30bnj535 31086 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3226, 31sylbi 207 . 2 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ∧ 𝑛 = suc 𝑚𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
3320, 32sylbi 207 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cun 3605  c0 3948  {csn 4210  cop 4216   ciun 4552  suc csuc 5763   Fn wfn 5921  cfv 5926  ωcom 7107  w-bnj17 30880   predc-bnj14 30882   FrSe w-bnj15 30886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-bnj17 30881  df-bnj14 30883  df-bnj13 30885  df-bnj15 30887
This theorem is referenced by:  bnj544  31090
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