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Theorem bnj1097 31656
Description: Technical lemma for bnj69 31685. 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 208 . . . . . . . 8 (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
41, 3bnj771 31441 . . . . . . 7 (𝜒 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
543ad2ant3 1126 . . . . . 6 ((𝜃𝜏𝜒) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
65adantr 474 . . . . 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 1138 . . . . . . 7 (𝜏 → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
983ad2ant2 1125 . . . . . 6 ((𝜃𝜏𝜒) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
109adantr 474 . . . . 5 (((𝜃𝜏𝜒) ∧ 𝜑0) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
116, 10jca 507 . . . 4 (((𝜃𝜏𝜒) ∧ 𝜑0) → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
1211anim2i 610 . . 3 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)))
13 3anass 1079 . . 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 226 . 2 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
15 fveqeq2 6457 . . . . . 6 (𝑖 = ∅ → ((𝑓𝑖) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
1615biimpar 471 . . . . 5 ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓𝑖) = pred(𝑋, 𝐴, 𝑅))
1716adantr 474 . . . 4 (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) = pred(𝑋, 𝐴, 𝑅))
18 simpr 479 . . . 4 (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
1917, 18eqsstrd 3858 . . 3 (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓𝑖) ⊆ 𝐵)
20193impa 1097 . 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 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  Vcvv 3398  wss 3792  c0 4141   Fn wfn 6132  cfv 6137  w-bnj17 31362   predc-bnj14 31364   TrFow-bnj19 31372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-bnj17 31363
This theorem is referenced by:  bnj1030  31662
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