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Theorem bnj540 31261
Description: Technical lemma for bnj852 31290. 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
bnj540.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj540.2 (𝜓″[𝐺 / 𝑓]𝜓)
bnj540.3 𝐺 ∈ V
Assertion
Ref Expression
bnj540 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐺,𝑖,𝑦   𝑓,𝑁   𝑅,𝑓
Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖)   𝐴(𝑦,𝑖)   𝑅(𝑦,𝑖)   𝑁(𝑦,𝑖)   𝜓″(𝑦,𝑓,𝑖)

Proof of Theorem bnj540
StepHypRef Expression
1 bnj540.2 . 2 (𝜓″[𝐺 / 𝑓]𝜓)
2 bnj540.1 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
32sbcbii 3624 . . 3 ([𝐺 / 𝑓]𝜓[𝐺 / 𝑓]𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 bnj540.3 . . . 4 𝐺 ∈ V
54bnj538 31108 . . 3 ([𝐺 / 𝑓]𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω [𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
6 sbcimg 3610 . . . . 5 (𝐺 ∈ V → ([𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝐺 / 𝑓]suc 𝑖𝑁[𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
74, 6ax-mp 5 . . . 4 ([𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝐺 / 𝑓]suc 𝑖𝑁[𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
87ralbii 3110 . . 3 (∀𝑖 ∈ ω [𝐺 / 𝑓](suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω ([𝐺 / 𝑓]suc 𝑖𝑁[𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
93, 5, 83bitri 286 . 2 ([𝐺 / 𝑓]𝜓 ↔ ∀𝑖 ∈ ω ([𝐺 / 𝑓]suc 𝑖𝑁[𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
104bnj525 31106 . . . 4 ([𝐺 / 𝑓]suc 𝑖𝑁 ↔ suc 𝑖𝑁)
11 fveq1 6343 . . . . . 6 (𝑓 = 𝐺 → (𝑓‘suc 𝑖) = (𝐺‘suc 𝑖))
12 fveq1 6343 . . . . . . 7 (𝑓 = 𝐺 → (𝑓𝑖) = (𝐺𝑖))
1312bnj1113 31155 . . . . . 6 (𝑓 = 𝐺 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
1411, 13eqeq12d 2767 . . . . 5 (𝑓 = 𝐺 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
154, 14sbcie 3603 . . . 4 ([𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
1610, 15imbi12i 339 . . 3 (([𝐺 / 𝑓]suc 𝑖𝑁[𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
1716ralbii 3110 . 2 (∀𝑖 ∈ ω ([𝐺 / 𝑓]suc 𝑖𝑁[𝐺 / 𝑓](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
181, 9, 173bitri 286 1 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  [wsbc 3568   ciun 4664  suc csuc 5878  cfv 6041  ωcom 7222   predc-bnj14 31055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-sbc 3569  df-in 3714  df-ss 3721  df-uni 4581  df-iun 4666  df-br 4797  df-iota 6004  df-fv 6049
This theorem is referenced by:  bnj580  31282  bnj607  31285
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