Theorem bnj1123 31828

Description: Technical lemma for bnj69 31852. 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 3752	. 2 ⊢ ([𝑗 / 𝑖]𝜂 ↔ [𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵))
4		bnj1123.3	. . . . . . . 8 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
5		nfcv 2947	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝐷
6		nfv 1890	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑓 Fn 𝑛
7		nfv 1890	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑
8		bnj1123.4	. . . . . . . . . . . . 13 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
9	8	bnj1095 31626	. . . . . . . . . . . 12 ⊢ (𝜓 → ∀𝑖𝜓)
10	9	nf5i 2115	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜓
11	6, 7, 10	nf3an 1881	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
12	5, 11	nfrex 3268	. . . . . . . . 9 ⊢ Ⅎ𝑖∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
13	12	nfab 2953	. . . . . . . 8 ⊢ Ⅎ𝑖{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
14	4, 13	nfcxfr 2945	. . . . . . 7 ⊢ Ⅎ𝑖𝐾
15	14	nfcri 2941	. . . . . 6 ⊢ Ⅎ𝑖 𝑓 ∈ 𝐾
16		nfv 1890	. . . . . 6 ⊢ Ⅎ𝑖 𝑗 ∈ dom 𝑓
17	15, 16	nfan 1879	. . . . 5 ⊢ Ⅎ𝑖(𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓)
18		nfv 1890	. . . . 5 ⊢ Ⅎ𝑖(𝑓‘𝑗) ⊆ 𝐵
19	17, 18	nfim 1876	. . . 4 ⊢ Ⅎ𝑖((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)
20		eleq1w 2863	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ dom 𝑓 ↔ 𝑗 ∈ dom 𝑓))
21	20	anbi2d 628	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓)))
22		fveq2 6530	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
23	22	sseq1d 3914	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑓‘𝑖) ⊆ 𝐵 ↔ (𝑓‘𝑗) ⊆ 𝐵))
24	21, 23	imbi12d 346	. . . 4 ⊢ (𝑖 = 𝑗 → (((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)))
25	19, 24	sbciegf 3733	. . 3 ⊢ (𝑗 ∈ V → ([𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)))
26	25	elv 3437	. 2 ⊢ ([𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
27	1, 3, 26	3bitri 298	1 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1078   = wceq 1520   ∈ wcel 2079  {cab 2773  ∀wral 3103  ∃wrex 3104  Vcvv 3432  [wsbc 3701   ⊆ wss 3854  ∪ ciun 4819  dom cdm 5435  suc csuc 6060   Fn wfn 6212  ‘cfv 6217  ωcom 7427   predc-bnj14 31531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-iota 6181  df-fv 6225

This theorem is referenced by:  bnj1030  31829