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Theorem bnj546 34888
Description: Technical lemma for bnj852 34913. 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 1151 . . . . . . 7 ((𝑓 Fn 𝑚𝜑′𝜓′) → (𝜑′𝜓′))
31, 2sylbi 217 . . . . . 6 (𝜏 → (𝜑′𝜓′))
4 bnj546.3 . . . . . . 7 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
5 bnj546.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
65bnj923 34760 . . . . . . . . 9 (𝑚𝐷𝑚 ∈ ω)
763ad2ant1 1134 . . . . . . . 8 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → 𝑚 ∈ ω)
8 simp3 1139 . . . . . . . 8 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → 𝑝𝑚)
97, 8jca 511 . . . . . . 7 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → (𝑚 ∈ ω ∧ 𝑝𝑚))
104, 9sylbi 217 . . . . . 6 (𝜎 → (𝑚 ∈ ω ∧ 𝑝𝑚))
113, 10anim12i 613 . . . . 5 ((𝜏𝜎) → ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
12 bnj256 34698 . . . . 5 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
1311, 12sylibr 234 . . . 4 ((𝜏𝜎) → (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
1413anim2i 617 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜏𝜎)) → (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)))
15143impb 1115 . 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 261 . . 3 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
1916, 17, 18bnj518 34878 . 2 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
20 fvex 6917 . . 3 (𝑓𝑝) ∈ V
21 iunexg 7984 . . 3 (((𝑓𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
2220, 21mpan 690 . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3060  Vcvv 3479  cdif 3947  c0 4332  {csn 4624   ciun 4989  suc csuc 6384   Fn wfn 6554  cfv 6559  ωcom 7883  w-bnj17 34678   predc-bnj14 34680   FrSe w-bnj15 34684
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 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751
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 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-tr 5258  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fv 6567  df-om 7884  df-bnj17 34679  df-bnj14 34681  df-bnj13 34683  df-bnj15 34685
This theorem is referenced by:  bnj938  34929
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