Theorem bnj1052 35372

Description: Technical lemma for bnj69 35407. 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 1972	. . . . 5 ⊢ (∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
10	9	albii 1848	. . . 4 ⊢ (∀𝑓∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ ∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
11		19.23v 1971	. . . 4 ⊢ (∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
12	10, 11	bitri 278	. . 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 3458	. . . . . . . . 9 ⊢ 𝑛 ∈ V
15		bnj1052.10	. . . . . . . . 9 ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
16	14, 15	bnj110 35255	. . . . . . . 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 35371	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝑛 𝜂 ↔ ∀𝑖𝜂)
19	16, 18	sylib 221	. . . . . . 7 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → ∀𝑖𝜂)
20	19	19.21bi 2224	. . . . . 6 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → 𝜂)
21	20, 17	sylib 221	. . . . 5 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
22	13, 21	mpcom 39	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
23	22	gen2 1825	. . 3 ⊢ ∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
24	12, 23	mpgbi 1827	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
25	1, 2, 3, 4, 5, 6, 7, 8, 24	bnj1034 35367	1 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102  ∀wal 1567   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740  ∀wral 3078  ∃wrex 3088  Vcvv 3454  [wsbc 3743   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  {csn 4588  ∪ ciun 4955   class class class wbr 5108   E cep 5559   Fr wfr 5610  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  ωcom 7860   ∧ w-bnj17 35084   predc-bnj14 35086   FrSe w-bnj15 35090   trClc-bnj18 35092   TrFow-bnj19 35094

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-iun 4957  df-br 5109  df-fr 5613  df-fn 6539  df-bnj17 35085  df-bnj18 35093

This theorem is used by:  bnj1053  35373