Theorem bnj1123 34028

Description: Technical lemma for bnj69 34052. 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 3838	. 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 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝐷
6		nfv 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑓 Fn 𝑛
7		nfv 1918	. . . . . . . . . . 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 33823	. . . . . . . . . . . 12 ⊢ (𝜓 → ∀𝑖𝜓)
10	9	nf5i 2143	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜓
11	6, 7, 10	nf3an 1905	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
12	5, 11	nfrexw 3311	. . . . . . . . 9 ⊢ Ⅎ𝑖∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
13	12	nfab 2910	. . . . . . . 8 ⊢ Ⅎ𝑖{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
14	4, 13	nfcxfr 2902	. . . . . . 7 ⊢ Ⅎ𝑖𝐾
15	14	nfcri 2891	. . . . . 6 ⊢ Ⅎ𝑖 𝑓 ∈ 𝐾
16		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑖 𝑗 ∈ dom 𝑓
17	15, 16	nfan 1903	. . . . 5 ⊢ Ⅎ𝑖(𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓)
18		nfv 1918	. . . . 5 ⊢ Ⅎ𝑖(𝑓‘𝑗) ⊆ 𝐵
19	17, 18	nfim 1900	. . . 4 ⊢ Ⅎ𝑖((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)
20		eleq1w 2817	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ dom 𝑓 ↔ 𝑗 ∈ dom 𝑓))
21	20	anbi2d 630	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓)))
22		fveq2 6892	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
23	22	sseq1d 4014	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑓‘𝑖) ⊆ 𝐵 ↔ (𝑓‘𝑗) ⊆ 𝐵))
24	21, 23	imbi12d 345	. . . 4 ⊢ (𝑖 = 𝑗 → (((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)))
25	19, 24	sbciegf 3817	. . 3 ⊢ (𝑗 ∈ V → ([𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)))
26	25	elv 3481	. 2 ⊢ ([𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
27	1, 3, 26	3bitri 297	1 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071  Vcvv 3475  [wsbc 3778   ⊆ wss 3949  ∪ ciun 4998  dom cdm 5677  suc csuc 6367   Fn wfn 6539  ‘cfv 6544  ωcom 7855   predc-bnj14 33730

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-3an 1090  df-tru 1545  df-fal 1555  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-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552

This theorem is referenced by:  bnj1030  34029