Theorem bnj1121 34997

Description: Technical lemma for bnj69 35022. 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 2184	. . . . 5 ⊢ (𝜒 → ∃𝑛𝜒)
2	1	bnj707 34767	. . . 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 34990	. . . 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 34768	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑖 ∈ 𝑛)
10	3	bnj1235 34816	. . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛)
11	10	bnj707 34767	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑓 Fn 𝑛)
12	11	fndmd 6586	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → dom 𝑓 = 𝑛)
13	9, 12	eleqtrrd 2834	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑖 ∈ dom 𝑓)
14		bnj1121.6	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂)
15	14, 9	bnj1294 34829	. . . 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 34768	. 2 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ (𝑓‘𝑖))
21	18, 20	sseldd 3930	1 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  dom cdm 5614   Fn wfn 6476  ‘cfv 6481   ∧ w-bnj17 34698   predc-bnj14 34700   FrSe w-bnj15 34704   TrFow-bnj19 34708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-ss 3914  df-fn 6484  df-bnj17 34699

This theorem is referenced by:  bnj1030  34999