Theorem bnj1030 31572

Description: Technical lemma for bnj69 31595. 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
bnj1030.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1030.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1030.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1030.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
bnj1030.5	⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1030.6	⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
bnj1030.7	⊢ 𝐷 = (ω ∖ {∅})
bnj1030.8	⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj1030.9	⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
bnj1030.10	⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
bnj1030.11	⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑)
bnj1030.12	⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓)
bnj1030.13	⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒)
bnj1030.14	⊢ (𝜃′ ↔ [𝑗 / 𝑖]𝜃)
bnj1030.15	⊢ (𝜏′ ↔ [𝑗 / 𝑖]𝜏)
bnj1030.16	⊢ (𝜁′ ↔ [𝑗 / 𝑖]𝜁)
bnj1030.17	⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂)
bnj1030.18	⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
bnj1030.19	⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))

Assertion

Ref	Expression
bnj1030	⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

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

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜃(𝑦)   𝜏(𝑦)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜁(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜎(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜌(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑗)   𝐷(𝑦,𝑧,𝑓,𝑛)   𝐾(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝑋(𝑗)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜃′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜏′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜁′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜑₀(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1030

Step	Hyp	Ref	Expression
1		bnj1030.1	. 2 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj1030.2	. 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj1030.3	. 2 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		bnj1030.4	. 2 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
5		bnj1030.5	. 2 ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
6		bnj1030.6	. 2 ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
7		bnj1030.7	. 2 ⊢ 𝐷 = (ω ∖ {∅})
8		bnj1030.8	. 2 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
9		19.23vv 2039	. . . . 5 ⊢ (∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
10	9	albii 1915	. . . 4 ⊢ (∀𝑓∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ ∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
11		19.23v 2038	. . . 4 ⊢ (∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
12	10, 11	bitri 267	. . 3 ⊢ (∀𝑓∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
13		bnj1030.9	. . . . 5 ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
14	7	bnj1071 31562	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
15	3, 14	bnj769 31349	. . . . . . 7 ⊢ (𝜒 → E Fr 𝑛)
16	15	bnj707 31342	. . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → E Fr 𝑛)
17		bnj1030.10	. . . . . . 7 ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
18		bnj1030.17	. . . . . . 7 ⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂)
19		bnj1030.18	. . . . . . 7 ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
20		bnj1030.19	. . . . . . 7 ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
21	2, 8, 13, 18	bnj1123 31571	. . . . . . . . . 10 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
22	2, 3, 5, 7, 19, 20, 21	bnj1118 31569	. . . . . . . . 9 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)
23	1, 3, 5	bnj1097 31566	. . . . . . . . 9 ⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)
24	22, 23	bnj1109 31374	. . . . . . . 8 ⊢ ∃𝑗(((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵)
25	24, 2, 3	bnj1093 31565	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))
26	13, 17, 18, 19, 20, 25	bnj1090 31564	. . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂))
27		vex 3388	. . . . . . 7 ⊢ 𝑛 ∈ V
28	27, 17	bnj110 31445	. . . . . 6 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → ∀𝑖 ∈ 𝑛 𝜂)
29	16, 26, 28	syl2anc 580	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂)
30	4, 5, 3, 6, 13, 29, 8	bnj1121 31570	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
31	30	gen2 1892	. . 3 ⊢ ∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
32	12, 31	mpgbi 1894	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
33	1, 2, 3, 4, 5, 6, 7, 8, 32	bnj1034 31555	1 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108  ∀wal 1651   = wceq 1653  ∃wex 1875   ∈ wcel 2157  {cab 2785  ∀wral 3089  ∃wrex 3090  Vcvv 3385  [wsbc 3633   ∖ cdif 3766   ⊆ wss 3769  ∅c0 4115  {csn 4368  ∪ ciun 4710   class class class wbr 4843   E cep 5224   Fr wfr 5268  dom cdm 5312  suc csuc 5943   Fn wfn 6096  ‘cfv 6101  ωcom 7299   ∧ w-bnj17 31272   predc-bnj14 31274   FrSe w-bnj15 31278   trClc-bnj18 31280   TrFow-bnj19 31282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fn 6104  df-fv 6109  df-om 7300  df-bnj17 31273  df-bnj18 31281  df-bnj19 31283

This theorem is referenced by:  bnj1124  31573