Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1097 Structured version   Visualization version   GIF version

Theorem bnj1097 34645
Description: Technical lemma for bnj69 34674. 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
bnj1097.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1097.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1097.5 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
Assertion
Ref Expression
bnj1097 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)

Proof of Theorem bnj1097
StepHypRef Expression
1 bnj1097.3 . . . . . . . 8 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
2 bnj1097.1 . . . . . . . . 9 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
32biimpi 215 . . . . . . . 8 (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
41, 3bnj771 34428 . . . . . . 7 (𝜒 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
543ad2ant3 1132 . . . . . 6 ((𝜃𝜏𝜒) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
65adantr 479 . . . . 5 (((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
7 bnj1097.5 . . . . . . . 8 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
87simp3bi 1144 . . . . . . 7 (𝜏 → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
983ad2ant2 1131 . . . . . 6 ((𝜃𝜏𝜒) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
109adantr 479 . . . . 5 (((𝜃𝜏𝜒) ∧ 𝜑0) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
116, 10jca 510 . . . 4 (((𝜃𝜏𝜒) ∧ 𝜑0) → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
1211anim2i 615 . . 3 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)))
13 3anass 1092 . . 3 ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) ↔ (𝑖 = ∅ ∧ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)))
1412, 13sylibr 233 . 2 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
15 fveqeq2 6911 . . . . . 6 (𝑖 = ∅ → ((𝑓𝑖) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
1615biimpar 476 . . . . 5 ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓𝑖) = pred(𝑋, 𝐴, 𝑅))
1716adantr 479 . . . 4 (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) = pred(𝑋, 𝐴, 𝑅))
18 simpr 483 . . . 4 (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
1917, 18eqsstrd 4020 . . 3 (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) ⊆ 𝐵)
20193impa 1107 . 2 ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) ⊆ 𝐵)
2114, 20syl 17 1 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3473  wss 3949  c0 4326   Fn wfn 6548  cfv 6553  w-bnj17 34350   predc-bnj14 34352   TrFow-bnj19 34360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-bnj17 34351
This theorem is referenced by:  bnj1030  34651
  Copyright terms: Public domain W3C validator