Theorem bnj540 30216

Description: Technical lemma for bnj852 30245. 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 3458	. . 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 30063	. . 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 3444	. . . . 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 2963	. . 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 285	. 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 30061	. . . 4 ⊢ ([𝐺 / 𝑓]suc 𝑖 ∈ 𝑁 ↔ suc 𝑖 ∈ 𝑁)
11		fveq1 6102	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑓‘suc 𝑖) = (𝐺‘suc 𝑖))
12		fveq1 6102	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑖) = (𝐺‘𝑖))
13	12	bnj1113 30110	. . . . . 6 ⊢ (𝑓 = 𝐺 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
14	11, 13	eqeq12d 2625	. . . . 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 3437	. . . 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 339	. . 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 2963	. 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 285	1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402  ∪ ciun 4455  suc csuc 5642  ‘cfv 5804  ωcom 6957   predc-bnj14 30007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-in 3547  df-ss 3554  df-uni 4373  df-iun 4457  df-br 4584  df-iota 5768  df-fv 5812

This theorem is referenced by:  bnj580  30237  bnj607  30240