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Theorem bnj229 34877
Description: Technical lemma for bnj517 34878. 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.)
Hypothesis
Ref Expression
bnj229.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj229 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) ⊆ 𝐴)
Distinct variable groups:   𝐴,𝑖,𝑚,𝑦   𝑖,𝐹,𝑚,𝑦   𝑖,𝑁,𝑚   𝑅,𝑖,𝑚
Allowed substitution hints:   𝜓(𝑦,𝑖,𝑚,𝑛)   𝐴(𝑛)   𝑅(𝑦,𝑛)   𝐹(𝑛)   𝑁(𝑦,𝑛)

Proof of Theorem bnj229
StepHypRef Expression
1 bnj213 34875 . . 3 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
21bnj226 34727 . 2 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
3 bnj229.1 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
43bnj222 34876 . . . . . . 7 (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
54bnj228 34728 . . . . . 6 ((𝑚 ∈ ω ∧ 𝜓) → (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
65adantl 481 . . . . 5 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
7 eleq1 2827 . . . . . . 7 (suc 𝑚 = 𝑛 → (suc 𝑚𝑁𝑛𝑁))
8 fveqeq2 6916 . . . . . . 7 (suc 𝑚 = 𝑛 → ((𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
97, 8imbi12d 344 . . . . . 6 (suc 𝑚 = 𝑛 → ((suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))))
109adantr 480 . . . . 5 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → ((suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))))
116, 10mpbid 232 . . . 4 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
12113impb 1114 . . 3 ((suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓) → (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
1312impcom 407 . 2 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))
142, 13bnj1262 34803 1 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963   ciun 4996  suc csuc 6388  cfv 6563  ωcom 7887   predc-bnj14 34681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-suc 6392  df-iota 6516  df-fv 6571  df-bnj14 34682
This theorem is referenced by: (None)
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