Theorem bnj1052 31169

Description: Technical lemma for bnj69 31204. 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
bnj1052.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1052.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1052.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1052.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
bnj1052.5	⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1052.6	⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
bnj1052.7	⊢ 𝐷 = (ω ∖ {∅})
bnj1052.8	⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj1052.9	⊢ (𝜂 ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
bnj1052.10	⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
bnj1052.37	⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)))

Assertion

Ref	Expression
bnj1052	⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝑧,𝐴,𝑓,𝑖,𝑛   𝐵,𝑓,𝑖,𝑛,𝑧   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑧,𝑅   𝑓,𝑋,𝑖,𝑛,𝑦   𝑧,𝑋   𝜂,𝑗   𝜏,𝑓,𝑖,𝑛,𝑧   𝜃,𝑓,𝑖,𝑛,𝑧   𝑖,𝑗,𝑛   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜃(𝑦,𝑗)   𝜏(𝑦,𝑗)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜁(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜌(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑗)   𝐵(𝑦,𝑗)   𝐷(𝑦,𝑧,𝑓,𝑗,𝑛)   𝑅(𝑗)   𝐾(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝑋(𝑗)

Proof of Theorem bnj1052

Step	Hyp	Ref	Expression
1		bnj1052.1	. 2 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj1052.2	. 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj1052.3	. 2 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		bnj1052.4	. 2 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
5		bnj1052.5	. 2 ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
6		bnj1052.6	. 2 ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
7		bnj1052.7	. 2 ⊢ 𝐷 = (ω ∖ {∅})
8		bnj1052.8	. 2 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
9		19.23vv 1912	. . . . 5 ⊢ (∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
10	9	albii 1787	. . . 4 ⊢ (∀𝑓∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ ∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
11		19.23v 1911	. . . 4 ⊢ (∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
12	10, 11	bitri 264	. . 3 ⊢ (∀𝑓∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
13		bnj1052.37	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)))
14		vex 3234	. . . . . . . . 9 ⊢ 𝑛 ∈ V
15		bnj1052.10	. . . . . . . . 9 ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
16	14, 15	bnj110 31054	. . . . . . . 8 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → ∀𝑖 ∈ 𝑛 𝜂)
17		bnj1052.9	. . . . . . . . 9 ⊢ (𝜂 ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
18	6, 17	bnj1049 31168	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝑛 𝜂 ↔ ∀𝑖𝜂)
19	16, 18	sylib 208	. . . . . . 7 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → ∀𝑖𝜂)
20	19	19.21bi 2097	. . . . . 6 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → 𝜂)
21	20, 17	sylib 208	. . . . 5 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
22	13, 21	mpcom 38	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
23	22	gen2 1763	. . 3 ⊢ ∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
24	12, 23	mpgbi 1765	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
25	1, 2, 3, 4, 5, 6, 7, 8, 24	bnj1034 31164	1 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942  Vcvv 3231  [wsbc 3468   ∖ cdif 3604   ⊆ wss 3607  ∅c0 3948  {csn 4210  ∪ ciun 4552   class class class wbr 4685   E cep 5057   Fr wfr 5099  suc csuc 5763   Fn wfn 5921  ‘cfv 5926  ωcom 7107   ∧ w-bnj17 30880   predc-bnj14 30882   FrSe w-bnj15 30886   trClc-bnj18 30888   TrFow-bnj19 30890

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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  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-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-iun 4554  df-br 4686  df-fr 5102  df-fn 5929  df-bnj17 30881  df-bnj18 30889

This theorem is referenced by:  bnj1053  31170