Theorem bnj1030 32976

Description: Technical lemma for bnj69 32999. 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 1950	. . . . 5 ⊢ (∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
10	9	albii 1826	. . . 4 ⊢ (∀𝑓∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ ∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
11		19.23v 1949	. . . 4 ⊢ (∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
12	10, 11	bitri 274	. . 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 32966	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
15	3, 14	bnj769 32751	. . . . . . 7 ⊢ (𝜒 → E Fr 𝑛)
16	15	bnj707 32744	. . . . . 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 32975	. . . . . . . . . 10 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
22	2, 3, 5, 7, 19, 20, 21	bnj1118 32973	. . . . . . . . 9 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)
23	1, 3, 5	bnj1097 32970	. . . . . . . . 9 ⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)
24	22, 23	bnj1109 32775	. . . . . . . 8 ⊢ ∃𝑗(((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵)
25	24, 2, 3	bnj1093 32969	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))
26	13, 17, 18, 19, 20, 25	bnj1090 32968	. . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂))
27		vex 3435	. . . . . . 7 ⊢ 𝑛 ∈ V
28	27, 17	bnj110 32847	. . . . . 6 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → ∀𝑖 ∈ 𝑛 𝜂)
29	16, 26, 28	syl2anc 584	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂)
30	4, 5, 3, 6, 13, 29, 8	bnj1121 32974	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
31	30	gen2 1803	. . 3 ⊢ ∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
32	12, 31	mpgbi 1805	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
33	1, 2, 3, 4, 5, 6, 7, 8, 32	bnj1034 32959	1 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1540   = wceq 1542  ∃wex 1786   ∈ wcel 2110  {cab 2717  ∀wral 3066  ∃wrex 3067  Vcvv 3431  [wsbc 3720   ∖ cdif 3889   ⊆ wss 3892  ∅c0 4262  {csn 4567  ∪ ciun 4930   class class class wbr 5079   E cep 5495   Fr wfr 5542  dom cdm 5590  suc csuc 6267   Fn wfn 6427  ‘cfv 6432  ωcom 7707   ∧ w-bnj17 32674   predc-bnj14 32676   FrSe w-bnj15 32680   trClc-bnj18 32682   TrFow-bnj19 32684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7583

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-tr 5197  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fn 6435  df-fv 6440  df-om 7708  df-bnj17 32675  df-bnj18 32683  df-bnj19 32685

This theorem is referenced by:  bnj1124  32977