Theorem bnj1021 32348

Description: Technical lemma for bnj69 32392. 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
bnj1021.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1021.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1021.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1021.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1021.5	⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
bnj1021.6	⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
bnj1021.13	⊢ 𝐷 = (ω ∖ {∅})
bnj1021.14	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}

Assertion

Ref	Expression
bnj1021	⊢ ∃𝑓∃𝑛∃𝑖∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑚,𝑝   𝜂,𝑚,𝑝   𝜃,𝑓,𝑖,𝑛   𝜑,𝑖   𝑚,𝑛,𝜃,𝑝

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝐴(𝑧,𝑚,𝑝)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝑅(𝑧,𝑚,𝑝)   𝑋(𝑧,𝑚,𝑝)

Proof of Theorem bnj1021

Step	Hyp	Ref	Expression
1		bnj1021.1	. . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj1021.2	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj1021.3	. . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		bnj1021.4	. . . 4 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5		bnj1021.5	. . . 4 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
6		bnj1021.6	. . . 4 ⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
7		bnj1021.13	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
8		bnj1021.14	. . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
9	1, 2, 3, 4, 5, 6, 7, 8	bnj996 32338	. . 3 ⊢ ∃𝑓∃𝑛∃𝑖∃𝑚∃𝑝(𝜃 → (𝜒 ∧ 𝜏 ∧ 𝜂))
10		anclb 549	. . . . . 6 ⊢ ((𝜃 → (𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒 ∧ 𝜏 ∧ 𝜂))))
11		bnj252 32083	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ (𝜃 ∧ (𝜒 ∧ 𝜏 ∧ 𝜂)))
12	11	imbi2i 339	. . . . . 6 ⊢ ((𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒 ∧ 𝜏 ∧ 𝜂))))
13	10, 12	bitr4i 281	. . . . 5 ⊢ ((𝜃 → (𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ (𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)))
14	13	2exbii 1850	. . . 4 ⊢ (∃𝑚∃𝑝(𝜃 → (𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ ∃𝑚∃𝑝(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)))
15	14	3exbii 1851	. . 3 ⊢ (∃𝑓∃𝑛∃𝑖∃𝑚∃𝑝(𝜃 → (𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ ∃𝑓∃𝑛∃𝑖∃𝑚∃𝑝(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)))
16	9, 15	mpbi 233	. 2 ⊢ ∃𝑓∃𝑛∃𝑖∃𝑚∃𝑝(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂))
17		19.37v 1998	. . . . 5 ⊢ (∃𝑝(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ (𝜃 → ∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)))
18		bnj1019 32161	. . . . . 6 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏))
19	18	imbi2i 339	. . . . 5 ⊢ ((𝜃 → ∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ (𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)))
20	17, 19	bitri 278	. . . 4 ⊢ (∃𝑝(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ (𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)))
21	20	2exbii 1850	. . 3 ⊢ (∃𝑖∃𝑚∃𝑝(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ ∃𝑖∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)))
22	21	2exbii 1850	. 2 ⊢ (∃𝑓∃𝑛∃𝑖∃𝑚∃𝑝(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂)) ↔ ∃𝑓∃𝑛∃𝑖∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)))
23	16, 22	mpbi 233	1 ⊢ ∃𝑓∃𝑛∃𝑖∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878  ∅c0 4243  {csn 4525  ∪ ciun 4881  suc csuc 6161   Fn wfn 6319  ‘cfv 6324  ωcom 7560   ∧ w-bnj17 32066   predc-bnj14 32068   FrSe w-bnj15 32072   trClc-bnj18 32074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-fn 6327  df-om 7561  df-bnj17 32067  df-bnj18 32075

This theorem is referenced by:  bnj907  32349