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Theorem bnj953 31337
Description: Technical lemma for bnj69 31406. 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
bnj953.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj953.2 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
Assertion
Ref Expression
bnj953 (((𝐺𝑖) = (𝑓𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))

Proof of Theorem bnj953
StepHypRef Expression
1 bnj312 31108 . . 3 (((𝐺𝑖) = (𝑓𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓) ↔ ((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝐺𝑖) = (𝑓𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓))
2 bnj251 31098 . . 3 (((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝐺𝑖) = (𝑓𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓) ↔ ((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ ((𝐺𝑖) = (𝑓𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓))))
31, 2bitri 264 . 2 (((𝐺𝑖) = (𝑓𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓) ↔ ((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ ((𝐺𝑖) = (𝑓𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓))))
4 bnj953.1 . . . . . 6 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
54bnj115 31121 . . . . 5 (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑛) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
6 sp 2200 . . . . . 6 (∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑛) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) → ((𝑖 ∈ ω ∧ suc 𝑖𝑛) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
76impcom 445 . . . . 5 (((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑛) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
85, 7sylan2b 493 . . . 4 (((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
9 bnj953.2 . . . . 5 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
109bnj956 31175 . . . 4 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
11 eqtr3 2781 . . . 4 (((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ∧ 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
128, 10, 11syl2anr 496 . . 3 (((𝐺𝑖) = (𝑓𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
13 eqtr 2779 . . 3 (((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑓‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
1412, 13sylan2 492 . 2 (((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ ((𝐺𝑖) = (𝑓𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓))) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
153, 14sylbi 207 1 (((𝐺𝑖) = (𝑓𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  wral 3050   ciun 4672  suc csuc 5886  cfv 6049  ωcom 7231  w-bnj17 31082   predc-bnj14 31084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-rex 3056  df-iun 4674  df-bnj17 31083
This theorem is referenced by:  bnj967  31343
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