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Theorem bnj557 30225
 Description: Technical lemma for bnj852 30245. 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
bnj557.3 𝐷 = (ω ∖ {∅})
bnj557.16 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj557.17 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj557.18 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj557.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj557.20 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
bnj557.21 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
bnj557.22 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
bnj557.23 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj557.24 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
bnj557.25 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj557.28 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj557.29 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj557.36 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Assertion
Ref Expression
bnj557 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺𝑚) = 𝐿)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑦,𝐺   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′
Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜁(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑚,𝑛)   𝐵(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑥,𝑓,𝑚,𝑛)   𝐺(𝑥,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐾(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐿(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 bnj557
StepHypRef Expression
1 3an4anass 1283 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)))
2 bnj557.18 . . . . . . . 8 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
3 bnj557.19 . . . . . . . 8 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
42, 3bnj556 30224 . . . . . . 7 (𝜂𝜎)
543anim3i 1243 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → (𝑅 FrSe 𝐴𝜏𝜎))
6 bnj557.20 . . . . . . 7 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
7 vex 3176 . . . . . . . 8 𝑖 ∈ V
87bnj216 30054 . . . . . . 7 (𝑚 = suc 𝑖𝑖𝑚)
96, 8bnj837 30085 . . . . . 6 (𝜁𝑖𝑚)
105, 9anim12i 588 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜂) ∧ 𝜁) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚))
111, 10sylbir 224 . . . 4 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚))
123bnj1254 30134 . . . . . 6 (𝜂𝑚 = suc 𝑝)
136simp3bi 1071 . . . . . 6 (𝜁𝑚 = suc 𝑖)
14 bnj551 30066 . . . . . 6 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
1512, 13, 14syl2an 493 . . . . 5 ((𝜂𝜁) → 𝑝 = 𝑖)
1615adantl 481 . . . 4 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → 𝑝 = 𝑖)
1711, 16jca 553 . . 3 (((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)) → (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) ∧ 𝑝 = 𝑖))
18 bnj256 30025 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) ↔ ((𝑅 FrSe 𝐴𝜏) ∧ (𝜂𝜁)))
19 df-3an 1033 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) ↔ (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) ∧ 𝑝 = 𝑖))
2017, 18, 193imtr4i 280 . 2 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖))
21 bnj557.28 . . 3 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
22 bnj557.29 . . 3 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
23 bnj557.3 . . 3 𝐷 = (ω ∖ {∅})
24 bnj557.16 . . 3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
25 bnj557.17 . . 3 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
26 bnj557.22 . . 3 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
27 bnj557.25 . . 3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
28 bnj557.21 . . 3 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
29 bnj557.23 . . 3 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
30 bnj557.24 . . 3 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
31 bnj557.36 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
3221, 22, 23, 24, 25, 2, 26, 27, 28, 29, 30, 31bnj553 30222 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐿)
3320, 32syl 17 1 ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺𝑚) = 𝐿)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537   ∪ cun 3538  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-eprel 4949  df-id 4953  df-fr 4997  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-bnj17 30006 This theorem is referenced by:  bnj558  30226
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