Theorem bnj540 33892

Description: Technical lemma for bnj852 33921. 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 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj540.2	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
bnj540.3	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
bnj540	⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) 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

Step	Hyp	Ref	Expression
1		bnj540.2	. 2 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
2		bnj540.1	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3	2	sbcbii 3837	. . 3 ⊢ ([𝐺 / 𝑓]𝜓 ↔ [𝐺 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4		bnj540.3	. . . 4 ⊢ 𝐺 ∈ V
5	4	bnj538 33740	. . 3 ⊢ ([𝐺 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω [𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
6		sbcimg 3828	. . . . 5 ⊢ (𝐺 ∈ V → ([𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝐺 / 𝑓]suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
7	4, 6	ax-mp 5	. . . 4 ⊢ ([𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝐺 / 𝑓]suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
8	7	ralbii 3094	. . 3 ⊢ (∀𝑖 ∈ ω [𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω ([𝐺 / 𝑓]suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
9	3, 5, 8	3bitri 297	. 2 ⊢ ([𝐺 / 𝑓]𝜓 ↔ ∀𝑖 ∈ ω ([𝐺 / 𝑓]suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
10	4	bnj525 33738	. . . 4 ⊢ ([𝐺 / 𝑓]suc 𝑖 ∈ 𝑁 ↔ suc 𝑖 ∈ 𝑁)
11		fveq1 6888	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑓‘suc 𝑖) = (𝐺‘suc 𝑖))
12		fveq1 6888	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑖) = (𝐺‘𝑖))
13	12	bnj1113 33785	. . . . . 6 ⊢ (𝑓 = 𝐺 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
14	11, 13	eqeq12d 2749	. . . . 5 ⊢ (𝑓 = 𝐺 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
15	4, 14	sbcie 3820	. . . 4 ⊢ ([𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
16	10, 15	imbi12i 351	. . 3 ⊢ (([𝐺 / 𝑓]suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
17	16	ralbii 3094	. 2 ⊢ (∀𝑖 ∈ ω ([𝐺 / 𝑓]suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
18	1, 9, 17	3bitri 297	1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  [wsbc 3777  ∪ ciun 4997  suc csuc 6364  ‘cfv 6541  ωcom 7852   predc-bnj14 33688

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-tru 1545  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-v 3477  df-sbc 3778  df-in 3955  df-ss 3965  df-uni 4909  df-iun 4999  df-br 5149  df-iota 6493  df-fv 6549

This theorem is referenced by:  bnj580  33913  bnj607  33916