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Theorem bnj229 32384
 Description: Technical lemma for bnj517 32385. 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 32382 . . 3 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
21bnj226 32232 . 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 32383 . . . . . . 7 (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
54bnj228 32233 . . . . . 6 ((𝑚 ∈ ω ∧ 𝜓) → (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
65adantl 485 . . . . 5 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
7 eleq1 2839 . . . . . . 7 (suc 𝑚 = 𝑛 → (suc 𝑚𝑁𝑛𝑁))
8 fveqeq2 6667 . . . . . . 7 (suc 𝑚 = 𝑛 → ((𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
97, 8imbi12d 348 . . . . . 6 (suc 𝑚 = 𝑛 → ((suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))))
109adantr 484 . . . . 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 235 . . . 4 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
12113impb 1112 . . 3 ((suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓) → (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
1312impcom 411 . 2 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))
142, 13bnj1262 32310 1 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) ⊆ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3858  ∪ ciun 4883  suc csuc 6171  ‘cfv 6335  ωcom 7579   predc-bnj14 32186 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-suc 6175  df-iota 6294  df-fv 6343  df-bnj14 32187 This theorem is referenced by: (None)
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