Theorem bnj1030 31055

Description: Technical lemma for bnj69 31078. 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 1903	. . . . 5 ⊢ (∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
10	9	albii 1747	. . . 4 ⊢ (∀𝑓∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ ∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
11		19.23v 1902	. . . 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		bnj1030.9	. . . . 5 ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
14	7	bnj1071 31045	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
15	3, 14	bnj769 30832	. . . . . . 7 ⊢ (𝜒 → E Fr 𝑛)
16	15	bnj707 30825	. . . . . 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 31054	. . . . . . . . . 10 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
22	2, 3, 5, 7, 19, 20, 21	bnj1118 31052	. . . . . . . . 9 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)
23	1, 3, 5	bnj1097 31049	. . . . . . . . 9 ⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)
24	22, 23	bnj1109 30857	. . . . . . . 8 ⊢ ∃𝑗(((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵)
25	24, 2, 3	bnj1093 31048	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))
26	13, 17, 18, 19, 20, 25	bnj1090 31047	. . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂))
27		vex 3203	. . . . . . 7 ⊢ 𝑛 ∈ V
28	27, 17	bnj110 30928	. . . . . 6 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → ∀𝑖 ∈ 𝑛 𝜂)
29	16, 26, 28	syl2anc 693	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂)
30	4, 5, 3, 6, 13, 29, 8	bnj1121 31053	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
31	30	gen2 1723	. . 3 ⊢ ∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
32	12, 31	mpgbi 1725	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
33	1, 2, 3, 4, 5, 6, 7, 8, 32	bnj1034 31038	1 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200  [wsbc 3435   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ ciun 4520   class class class wbr 4653   E cep 5028   Fr wfr 5070  dom cdm 5114  suc csuc 5725   Fn wfn 5883  ‘cfv 5888  ωcom 7065   ∧ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760   TrFow-bnj19 30762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1124  31056