Theorem bnj1121 34999

Description: Technical lemma for bnj69 35024. 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
bnj1121.1	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
bnj1121.2	⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1121.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1121.4	⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
bnj1121.5	⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
bnj1121.6	⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂)
bnj1121.7	⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}

Assertion

Ref	Expression
bnj1121	⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)

Proof of Theorem bnj1121

Step	Hyp	Ref	Expression
1		19.8a 2181	. . . . 5 ⊢ (𝜒 → ∃𝑛𝜒)
2	1	bnj707 34769	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∃𝑛𝜒)
3		bnj1121.3	. . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		bnj1121.7	. . . . 5 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
5	3, 4	bnj1083 34992	. . . 4 ⊢ (𝑓 ∈ 𝐾 ↔ ∃𝑛𝜒)
6	2, 5	sylibr 234	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑓 ∈ 𝐾)
7		bnj1121.4	. . . . . 6 ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
8	7	simplbi 497	. . . . 5 ⊢ (𝜁 → 𝑖 ∈ 𝑛)
9	8	bnj708 34770	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑖 ∈ 𝑛)
10	3	bnj1235 34818	. . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛)
11	10	bnj707 34769	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑓 Fn 𝑛)
12	11	fndmd 6673	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → dom 𝑓 = 𝑛)
13	9, 12	eleqtrrd 2844	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑖 ∈ dom 𝑓)
14		bnj1121.6	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂)
15	14, 9	bnj1294 34831	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝜂)
16		bnj1121.5	. . . 4 ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
17	15, 16	sylib 218	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
18	6, 13, 17	mp2and 699	. 2 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → (𝑓‘𝑖) ⊆ 𝐵)
19	7	simprbi 496	. . 3 ⊢ (𝜁 → 𝑧 ∈ (𝑓‘𝑖))
20	19	bnj708 34770	. 2 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ (𝑓‘𝑖))
21	18, 20	sseldd 3984	1 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  dom cdm 5685   Fn wfn 6556  ‘cfv 6561   ∧ w-bnj17 34700   predc-bnj14 34702   FrSe w-bnj15 34706   TrFow-bnj19 34710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-ss 3968  df-fn 6564  df-bnj17 34701

This theorem is referenced by:  bnj1030  35001