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Theorem bnj1097 33650
Description: Technical lemma for bnj69 33679. 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 33433 . . . . . . 7 (𝜒 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
543ad2ant3 1136 . . . . . 6 ((𝜃𝜏𝜒) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
65adantr 482 . . . . 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 1148 . . . . . . 7 (𝜏 → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
983ad2ant2 1135 . . . . . 6 ((𝜃𝜏𝜒) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
109adantr 482 . . . . 5 (((𝜃𝜏𝜒) ∧ 𝜑0) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
116, 10jca 513 . . . 4 (((𝜃𝜏𝜒) ∧ 𝜑0) → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
1211anim2i 618 . . 3 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)))
13 3anass 1096 . . 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 6852 . . . . . 6 (𝑖 = ∅ → ((𝑓𝑖) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
1615biimpar 479 . . . . 5 ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓𝑖) = pred(𝑋, 𝐴, 𝑅))
1716adantr 482 . . . 4 (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) = pred(𝑋, 𝐴, 𝑅))
18 simpr 486 . . . 4 (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
1917, 18eqsstrd 3983 . . 3 (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) ⊆ 𝐵)
20193impa 1111 . 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 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3444  wss 3911  c0 4283   Fn wfn 6492  cfv 6497  w-bnj17 33355   predc-bnj14 33357   TrFow-bnj19 33365
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-bnj17 33356
This theorem is referenced by:  bnj1030  33656
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