Theorem bnj611 30988

Description: Technical lemma for bnj852 30991. 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
bnj611.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj611.2	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
bnj611.3	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
bnj611	⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑓   𝑖,𝐺,𝑦   𝑓,𝑁   𝑅,𝑓   𝑓,𝑖,𝑦

Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖)   𝐴(𝑦,𝑖)   𝑅(𝑦,𝑖)   𝐺(𝑓)   𝑁(𝑦,𝑖)   𝜓″(𝑦,𝑓,𝑖)

Proof of Theorem bnj611

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj611.2	. 2 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
2		df-ral 2917	. . . . 5 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
3	2	bicomi 214	. . . 4 ⊢ (∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4	3	sbcbii 3491	. . 3 ⊢ ([𝐺 / 𝑓]∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [𝐺 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
5		bnj611.3	. . . . . . 7 ⊢ 𝐺 ∈ V
6		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑖 ∈ ω
7	6	sbc19.21g 3502	. . . . . . 7 ⊢ (𝐺 ∈ V → ([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)))))
8	5, 7	ax-mp 5	. . . . . 6 ⊢ ([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))))
9		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑓 suc 𝑖 ∈ 𝑁
10	9	sbc19.21g 3502	. . . . . . . . 9 ⊢ (𝐺 ∈ V → ([𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
11	5, 10	ax-mp 5	. . . . . . . 8 ⊢ ([𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑁 → [𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
12		fveq1 6190	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → (𝑓‘suc 𝑖) = (𝐺‘suc 𝑖))
13		fveq1 6190	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑖) = (𝐺‘𝑖))
14	13	bnj1113 30856	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
15	12, 14	eqeq12d 2637	. . . . . . . . . 10 ⊢ (𝑓 = 𝐺 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
16		fveq1 6190	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑒 → (𝑓‘suc 𝑖) = (𝑒‘suc 𝑖))
17		fveq1 6190	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑒 → (𝑓‘𝑖) = (𝑒‘𝑖))
18	17	bnj1113 30856	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑒 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅))
19	16, 18	eqeq12d 2637	. . . . . . . . . 10 ⊢ (𝑓 = 𝑒 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑒‘suc 𝑖) = ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅)))
20		fveq1 6190	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐺 → (𝑒‘suc 𝑖) = (𝐺‘suc 𝑖))
21		fveq1 6190	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝐺 → (𝑒‘𝑖) = (𝐺‘𝑖))
22	21	bnj1113 30856	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐺 → ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
23	20, 22	eqeq12d 2637	. . . . . . . . . 10 ⊢ (𝑒 = 𝐺 → ((𝑒‘suc 𝑖) = ∪ 𝑦 ∈ (𝑒‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
24	5, 15, 19, 23	bnj610 30817	. . . . . . . . 9 ⊢ ([𝐺 / 𝑓](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
25	24	imbi2i 326	. . . . . . . 8 ⊢ ((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)))
26	11, 25	bitri 264	. . . . . . 7 ⊢ ([𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
27	26	imbi2i 326	. . . . . 6 ⊢ ((𝑖 ∈ ω → [𝐺 / 𝑓](suc 𝑖 ∈ 𝑁 → (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))))
28	8, 27	bitri 264	. . . . 5 ⊢ ([𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (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))))
29	28	albii 1747	. . . 4 ⊢ (∀𝑖[𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
30		sbcal 3485	. . . 4 ⊢ ([𝐺 / 𝑓]∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ∀𝑖[𝐺 / 𝑓](𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
31		df-ral 2917	. . . 4 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))))
32	29, 30, 31	3bitr4ri 293	. . 3 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝐺 / 𝑓]∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
33		bnj611.1	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
34	33	sbcbii 3491	. . 3 ⊢ ([𝐺 / 𝑓]𝜓 ↔ [𝐺 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
35	4, 32, 34	3bitr4ri 293	. 2 ⊢ ([𝐺 / 𝑓]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
36	1, 35	bitri 264	1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  [wsbc 3435  ∪ ciun 4520  suc csuc 5725  ‘cfv 5888  ωcom 7065   predc-bnj14 30754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-in 3581  df-ss 3588  df-uni 4437  df-iun 4522  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj600  30989  bnj908  31001