Theorem bnj1123 31371

Description: Technical lemma for bnj69 31395. 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
bnj1123.4	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1123.3	⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj1123.1	⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
bnj1123.2	⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂)

Assertion

Ref	Expression
bnj1123	⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))

Distinct variable groups:   𝐵,𝑖   𝐷,𝑖   𝑓,𝑖   𝑖,𝑗   𝑖,𝑛   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜂(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑦,𝑓,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑗,𝑛)   𝑅(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐾(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑦,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1123

Step	Hyp	Ref	Expression
1		bnj1123.2	. 2 ⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂)
2		bnj1123.1	. . 3 ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
3	2	sbcbii 3683	. 2 ⊢ ([𝑗 / 𝑖]𝜂 ↔ [𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
4		vex 3390	. . 3 ⊢ 𝑗 ∈ V
5		bnj1123.3	. . . . . . . 8 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
6		nfcv 2944	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝐷
7		nfv 2005	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑓 Fn 𝑛
8		nfv 2005	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑
9		bnj1123.4	. . . . . . . . . . . . 13 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
10	9	bnj1095 31169	. . . . . . . . . . . 12 ⊢ (𝜓 → ∀𝑖𝜓)
11	10	nf5i 2189	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜓
12	7, 8, 11	nf3an 1993	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
13	6, 12	nfrex 3190	. . . . . . . . 9 ⊢ Ⅎ𝑖∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
14	13	nfab 2949	. . . . . . . 8 ⊢ Ⅎ𝑖{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
15	5, 14	nfcxfr 2942	. . . . . . 7 ⊢ Ⅎ𝑖𝐾
16	15	nfcri 2938	. . . . . 6 ⊢ Ⅎ𝑖 𝑓 ∈ 𝐾
17		nfv 2005	. . . . . 6 ⊢ Ⅎ𝑖 𝑗 ∈ dom 𝑓
18	16, 17	nfan 1990	. . . . 5 ⊢ Ⅎ𝑖(𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓)
19		nfv 2005	. . . . 5 ⊢ Ⅎ𝑖(𝑓‘𝑗) ⊆ 𝐵
20	18, 19	nfim 1987	. . . 4 ⊢ Ⅎ𝑖((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)
21		eleq1w 2864	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ dom 𝑓 ↔ 𝑗 ∈ dom 𝑓))
22	21	anbi2d 616	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓)))
23		fveq2 6402	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
24	23	sseq1d 3823	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑓‘𝑖) ⊆ 𝐵 ↔ (𝑓‘𝑗) ⊆ 𝐵))
25	22, 24	imbi12d 335	. . . 4 ⊢ (𝑖 = 𝑗 → (((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)))
26	20, 25	sbciegf 3659	. . 3 ⊢ (𝑗 ∈ V → ([𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)))
27	4, 26	ax-mp 5	. 2 ⊢ ([𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
28	1, 3, 27	3bitri 288	1 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1100   = wceq 1637   ∈ wcel 2155  {cab 2788  ∀wral 3092  ∃wrex 3093  Vcvv 3387  [wsbc 3627   ⊆ wss 3763  ∪ ciun 4705  dom cdm 5305  suc csuc 5932   Fn wfn 6090  ‘cfv 6095  ωcom 7289   predc-bnj14 31073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-br 4838  df-iota 6058  df-fv 6103

This theorem is referenced by:  bnj1030  31372