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 31863
Description: Technical lemma for bnj69 31888. 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 2143 . . . . 5 (𝜒 → ∃𝑛𝜒)
21bnj707 31635 . . . 4 ((𝜃𝜏𝜒𝜁) → ∃𝑛𝜒)
3 bnj1121.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1121.7 . . . . 5 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj1083 31856 . . . 4 (𝑓𝐾 ↔ ∃𝑛𝜒)
62, 5sylibr 235 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑓𝐾)
7 bnj1121.4 . . . . . 6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
87simplbi 498 . . . . 5 (𝜁𝑖𝑛)
98bnj708 31636 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑖𝑛)
103bnj1235 31685 . . . . . 6 (𝜒𝑓 Fn 𝑛)
1110bnj707 31635 . . . . 5 ((𝜃𝜏𝜒𝜁) → 𝑓 Fn 𝑛)
1211fndmd 6329 . . . 4 ((𝜃𝜏𝜒𝜁) → dom 𝑓 = 𝑛)
139, 12eleqtrrd 2885 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑖 ∈ dom 𝑓)
14 bnj1121.6 . . . . 5 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
1514, 9bnj1294 31698 . . . 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 31636 . 2 ((𝜃𝜏𝜒𝜁) → 𝑧 ∈ (𝑓𝑖))
2118, 20sseldd 3892 1 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wex 1762  wcel 2080  {cab 2774  wral 3104  wrex 3105  Vcvv 3436  wss 3861  dom cdm 5446   Fn wfn 6223  cfv 6228  w-bnj17 31565   predc-bnj14 31567   FrSe w-bnj15 31571   TrFow-bnj19 31575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-ext 2768
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-ral 3109  df-rex 3110  df-in 3868  df-ss 3876  df-fn 6231  df-bnj17 31566
This theorem is referenced by:  bnj1030  31865
  Copyright terms: Public domain W3C validator