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Theorem bnj546 31092
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
bnj546.1 𝐷 = (ω ∖ {∅})
bnj546.2 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj546.3 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj546.4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj546.5 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj546 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
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)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑥,𝑓,𝑚,𝑛)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj546
StepHypRef Expression
1 bnj546.2 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
2 3simpc 1080 . . . . . . 7 ((𝑓 Fn 𝑚𝜑′𝜓′) → (𝜑′𝜓′))
31, 2sylbi 207 . . . . . 6 (𝜏 → (𝜑′𝜓′))
4 bnj546.3 . . . . . . 7 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
5 bnj546.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
65bnj923 30964 . . . . . . . . 9 (𝑚𝐷𝑚 ∈ ω)
763ad2ant1 1102 . . . . . . . 8 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → 𝑚 ∈ ω)
8 simp3 1083 . . . . . . . 8 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → 𝑝𝑚)
97, 8jca 553 . . . . . . 7 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → (𝑚 ∈ ω ∧ 𝑝𝑚))
104, 9sylbi 207 . . . . . 6 (𝜎 → (𝑚 ∈ ω ∧ 𝑝𝑚))
113, 10anim12i 589 . . . . 5 ((𝜏𝜎) → ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
12 bnj256 30900 . . . . 5 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
1311, 12sylibr 224 . . . 4 ((𝜏𝜎) → (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
1413anim2i 592 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜏𝜎)) → (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)))
15143impb 1279 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)))
16 bnj546.4 . . 3 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
17 bnj546.5 . . 3 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
18 biid 251 . . 3 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
1916, 17, 18bnj518 31082 . 2 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
20 fvex 6239 . . 3 (𝑓𝑝) ∈ V
21 iunexg 7185 . . 3 (((𝑓𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
2220, 21mpan 706 . 2 (∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
2315, 19, 223syl 18 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  c0 3948  {csn 4210   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
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:  bnj938  31133
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