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Theorem bnj546 32589
Description: Technical lemma for bnj852 32614. 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 1152 . . . . . . 7 ((𝑓 Fn 𝑚𝜑′𝜓′) → (𝜑′𝜓′))
31, 2sylbi 220 . . . . . 6 (𝜏 → (𝜑′𝜓′))
4 bnj546.3 . . . . . . 7 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
5 bnj546.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
65bnj923 32460 . . . . . . . . 9 (𝑚𝐷𝑚 ∈ ω)
763ad2ant1 1135 . . . . . . . 8 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → 𝑚 ∈ ω)
8 simp3 1140 . . . . . . . 8 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → 𝑝𝑚)
97, 8jca 515 . . . . . . 7 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) → (𝑚 ∈ ω ∧ 𝑝𝑚))
104, 9sylbi 220 . . . . . 6 (𝜎 → (𝑚 ∈ ω ∧ 𝑝𝑚))
113, 10anim12i 616 . . . . 5 ((𝜏𝜎) → ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
12 bnj256 32397 . . . . 5 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ ((𝜑′𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝𝑚)))
1311, 12sylibr 237 . . . 4 ((𝜏𝜎) → (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
1413anim2i 620 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝜏𝜎)) → (𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)))
15143impb 1117 . 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 264 . . 3 ((𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚) ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))
1916, 17, 18bnj518 32579 . 2 ((𝑅 FrSe 𝐴 ∧ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚)) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
20 fvex 6730 . . 3 (𝑓𝑝) ∈ V
21 iunexg 7736 . . 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  Vcvv 3408  cdif 3863  c0 4237  {csn 4541   ciun 4904  suc csuc 6215   Fn wfn 6375  cfv 6380  ωcom 7644  w-bnj17 32377   predc-bnj14 32379   FrSe w-bnj15 32383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-bnj17 32378  df-bnj14 32380  df-bnj13 32382  df-bnj15 32384
This theorem is referenced by:  bnj938  32630
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