Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1121 Structured version   Visualization version   GIF version

Theorem bnj1121 31570
Description: Technical lemma for bnj69 31595. 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 2216 . . . . 5 (𝜒 → ∃𝑛𝜒)
21bnj707 31342 . . . 4 ((𝜃𝜏𝜒𝜁) → ∃𝑛𝜒)
3 bnj1121.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1121.7 . . . . 5 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj1083 31563 . . . 4 (𝑓𝐾 ↔ ∃𝑛𝜒)
62, 5sylibr 226 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑓𝐾)
7 bnj1121.4 . . . . . 6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
87simplbi 492 . . . . 5 (𝜁𝑖𝑛)
98bnj708 31343 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑖𝑛)
103bnj1235 31392 . . . . . 6 (𝜒𝑓 Fn 𝑛)
1110bnj707 31342 . . . . 5 ((𝜃𝜏𝜒𝜁) → 𝑓 Fn 𝑛)
12 fndm 6201 . . . . 5 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
1311, 12syl 17 . . . 4 ((𝜃𝜏𝜒𝜁) → dom 𝑓 = 𝑛)
149, 13eleqtrrd 2881 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑖 ∈ dom 𝑓)
15 bnj1121.6 . . . . 5 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
1615, 9bnj1294 31405 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝜂)
17 bnj1121.5 . . . 4 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
1816, 17sylib 210 . . 3 ((𝜃𝜏𝜒𝜁) → ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
196, 14, 18mp2and 691 . 2 ((𝜃𝜏𝜒𝜁) → (𝑓𝑖) ⊆ 𝐵)
207simprbi 491 . . 3 (𝜁𝑧 ∈ (𝑓𝑖))
2120bnj708 31343 . 2 ((𝜃𝜏𝜒𝜁) → 𝑧 ∈ (𝑓𝑖))
2219, 21sseldd 3799 1 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  {cab 2785  wral 3089  wrex 3090  Vcvv 3385  wss 3769  dom cdm 5312   Fn wfn 6096  cfv 6101  w-bnj17 31272   predc-bnj14 31274   FrSe w-bnj15 31278   TrFow-bnj19 31282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-ral 3094  df-rex 3095  df-in 3776  df-ss 3783  df-fn 6104  df-bnj17 31273
This theorem is referenced by:  bnj1030  31572
  Copyright terms: Public domain W3C validator