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Theorem bnj229 33895
Description: Technical lemma for bnj517 33896. 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 33893 . . 3 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
21bnj226 33745 . 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 33894 . . . . . . 7 (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
54bnj228 33746 . . . . . 6 ((𝑚 ∈ ω ∧ 𝜓) → (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
65adantl 483 . . . . 5 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
7 eleq1 2822 . . . . . . 7 (suc 𝑚 = 𝑛 → (suc 𝑚𝑁𝑛𝑁))
8 fveqeq2 6901 . . . . . . 7 (suc 𝑚 = 𝑛 → ((𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
97, 8imbi12d 345 . . . . . 6 (suc 𝑚 = 𝑛 → ((suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))))
109adantr 482 . . . . 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 231 . . . 4 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
12113impb 1116 . . 3 ((suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓) → (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
1312impcom 409 . 2 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))
142, 13bnj1262 33821 1 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wss 3949   ciun 4998  suc csuc 6367  cfv 6544  ωcom 7855   predc-bnj14 33699
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-10 2138  ax-11 2155  ax-12 2172  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-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-suc 6371  df-iota 6496  df-fv 6552  df-bnj14 33700
This theorem is referenced by: (None)
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