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Theorem bnj1121 32154
Description: Technical lemma for bnj69 32179. 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 𝐴𝑋𝐴))
bnj1121.2 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1121.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1121.4 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
bnj1121.5 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
bnj1121.6 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
bnj1121.7 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj1121 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)

Proof of Theorem bnj1121
StepHypRef Expression
1 19.8a 2170 . . . . 5 (𝜒 → ∃𝑛𝜒)
21bnj707 31925 . . . 4 ((𝜃𝜏𝜒𝜁) → ∃𝑛𝜒)
3 bnj1121.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1121.7 . . . . 5 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj1083 32147 . . . 4 (𝑓𝐾 ↔ ∃𝑛𝜒)
62, 5sylibr 235 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑓𝐾)
7 bnj1121.4 . . . . . 6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
87simplbi 498 . . . . 5 (𝜁𝑖𝑛)
98bnj708 31926 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑖𝑛)
103bnj1235 31975 . . . . . 6 (𝜒𝑓 Fn 𝑛)
1110bnj707 31925 . . . . 5 ((𝜃𝜏𝜒𝜁) → 𝑓 Fn 𝑛)
1211fndmd 6449 . . . 4 ((𝜃𝜏𝜒𝜁) → dom 𝑓 = 𝑛)
139, 12eleqtrrd 2913 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑖 ∈ dom 𝑓)
14 bnj1121.6 . . . . 5 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
1514, 9bnj1294 31988 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝜂)
16 bnj1121.5 . . . 4 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
1715, 16sylib 219 . . 3 ((𝜃𝜏𝜒𝜁) → ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
186, 13, 17mp2and 695 . 2 ((𝜃𝜏𝜒𝜁) → (𝑓𝑖) ⊆ 𝐵)
197simprbi 497 . . 3 (𝜁𝑧 ∈ (𝑓𝑖))
2019bnj708 31926 . 2 ((𝜃𝜏𝜒𝜁) → 𝑧 ∈ (𝑓𝑖))
2118, 20sseldd 3965 1 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  Vcvv 3492  wss 3933  dom cdm 5548   Fn wfn 6343  cfv 6348  w-bnj17 31855   predc-bnj14 31857   FrSe w-bnj15 31861   TrFow-bnj19 31865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ral 3140  df-rex 3141  df-in 3940  df-ss 3949  df-fn 6351  df-bnj17 31856
This theorem is referenced by:  bnj1030  32156
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