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Theorem bnj1121 33996
Description: Technical lemma for bnj69 34021. 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 2175 . . . . 5 (𝜒 → ∃𝑛𝜒)
21bnj707 33766 . . . 4 ((𝜃𝜏𝜒𝜁) → ∃𝑛𝜒)
3 bnj1121.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1121.7 . . . . 5 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj1083 33989 . . . 4 (𝑓𝐾 ↔ ∃𝑛𝜒)
62, 5sylibr 233 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑓𝐾)
7 bnj1121.4 . . . . . 6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
87simplbi 499 . . . . 5 (𝜁𝑖𝑛)
98bnj708 33767 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑖𝑛)
103bnj1235 33815 . . . . . 6 (𝜒𝑓 Fn 𝑛)
1110bnj707 33766 . . . . 5 ((𝜃𝜏𝜒𝜁) → 𝑓 Fn 𝑛)
1211fndmd 6655 . . . 4 ((𝜃𝜏𝜒𝜁) → dom 𝑓 = 𝑛)
139, 12eleqtrrd 2837 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑖 ∈ dom 𝑓)
14 bnj1121.6 . . . . 5 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
1514, 9bnj1294 33828 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝜂)
16 bnj1121.5 . . . 4 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
1715, 16sylib 217 . . 3 ((𝜃𝜏𝜒𝜁) → ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
186, 13, 17mp2and 698 . 2 ((𝜃𝜏𝜒𝜁) → (𝑓𝑖) ⊆ 𝐵)
197simprbi 498 . . 3 (𝜁𝑧 ∈ (𝑓𝑖))
2019bnj708 33767 . 2 ((𝜃𝜏𝜒𝜁) → 𝑧 ∈ (𝑓𝑖))
2118, 20sseldd 3984 1 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3062  wrex 3071  Vcvv 3475  wss 3949  dom cdm 5677   Fn wfn 6539  cfv 6544  w-bnj17 33697   predc-bnj14 33699   FrSe w-bnj15 33703   TrFow-bnj19 33707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-in 3956  df-ss 3966  df-fn 6547  df-bnj17 33698
This theorem is referenced by:  bnj1030  33998
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