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Theorem bnj953 32490
Description: Technical lemma for bnj69 32561. 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 32261 . . 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 32251 . . 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 278 . 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 32274 . . . . 5 (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑛) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
6 sp 2184 . . . . . 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 411 . . . . 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 597 . . . 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 32327 . . . 4 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
11 eqtr3 2760 . . . 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 600 . . 3 (((𝐺𝑖) = (𝑓𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓)) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
13 eqtr 2758 . . 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 596 . 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 220 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 209  wa 399  wal 1540   = wceq 1542  wcel 2114  wral 3053   ciun 4881  suc csuc 6174  cfv 6339  ωcom 7599  w-bnj17 32235   predc-bnj14 32237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1090  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-iun 4883  df-bnj17 32236
This theorem is referenced by:  bnj967  32496
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