Theorem bnj1014 32841

Description: Technical lemma for bnj69 32890. 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
bnj1014.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1014.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1014.13	⊢ 𝐷 = (ω ∖ {∅})
bnj1014.14	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}

Assertion

Ref	Expression
bnj1014	⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝑓,𝑔,𝑖   𝑖,𝑗   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑔,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝐴(𝑔,𝑗)   𝐵(𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑔,𝑗,𝑛)   𝑅(𝑔,𝑗)   𝑋(𝑔,𝑗)

Proof of Theorem bnj1014

Step	Hyp	Ref	Expression
1		bnj1014.14	. . . . . . 7 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
2		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑖𝐷
3		bnj1014.1	. . . . . . . . . . 11 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
4		bnj1014.2	. . . . . . . . . . 11 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
5	3, 4	bnj911 32812	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	5	nf5i 2144	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
7	2, 6	nfrex 3237	. . . . . . . 8 ⊢ Ⅎ𝑖∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
8	7	nfab 2912	. . . . . . 7 ⊢ Ⅎ𝑖{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
9	1, 8	nfcxfr 2904	. . . . . 6 ⊢ Ⅎ𝑖𝐵
10	9	nfcri 2893	. . . . 5 ⊢ Ⅎ𝑖 𝑔 ∈ 𝐵
11		nfv 1918	. . . . 5 ⊢ Ⅎ𝑖 𝑗 ∈ dom 𝑔
12	10, 11	nfan 1903	. . . 4 ⊢ Ⅎ𝑖(𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔)
13		nfv 1918	. . . 4 ⊢ Ⅎ𝑖(𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)
14	12, 13	nfim 1900	. . 3 ⊢ Ⅎ𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
15	14	nf5ri 2191	. 2 ⊢ (((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) → ∀𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))
16		eleq1w 2821	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ dom 𝑔 ↔ 𝑖 ∈ dom 𝑔))
17	16	anbi2d 628	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) ↔ (𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)))
18		fveq2 6756	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑔‘𝑗) = (𝑔‘𝑖))
19	18	sseq1d 3948	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
20	17, 19	imbi12d 344	. . . 4 ⊢ (𝑗 = 𝑖 → (((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
21	20	equcoms 2024	. . 3 ⊢ (𝑖 = 𝑗 → (((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
22	1	bnj1317 32701	. . . . . . 7 ⊢ (𝑔 ∈ 𝐵 → ∀𝑓 𝑔 ∈ 𝐵)
23	22	nf5i 2144	. . . . . 6 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐵
24		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑓 𝑖 ∈ dom 𝑔
25	23, 24	nfan 1903	. . . . 5 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)
26		nfv 1918	. . . . 5 ⊢ Ⅎ𝑓(𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)
27	25, 26	nfim 1900	. . . 4 ⊢ Ⅎ𝑓((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
28		eleq1w 2821	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐵 ↔ 𝑔 ∈ 𝐵))
29		dmeq 5801	. . . . . . 7 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
30	29	eleq2d 2824	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ dom 𝑔))
31	28, 30	anbi12d 630	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)))
32		fveq1 6755	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑖) = (𝑔‘𝑖))
33	32	sseq1d 3948	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
34	31, 33	imbi12d 344	. . . 4 ⊢ (𝑓 = 𝑔 → (((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))))
35		ssiun2 4973	. . . . 5 ⊢ (𝑖 ∈ dom 𝑓 → (𝑓‘𝑖) ⊆ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
36		ssiun2 4973	. . . . . 6 ⊢ (𝑓 ∈ 𝐵 → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
37		bnj1014.13	. . . . . . 7 ⊢ 𝐷 = (ω ∖ {∅})
38	3, 4, 37, 1	bnj882 32806	. . . . . 6 ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
39	36, 38	sseqtrrdi 3968	. . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
40	35, 39	sylan9ssr 3931	. . . 4 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
41	27, 34, 40	chvarfv 2236	. . 3 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
42	21, 41	speivw 1978	. 2 ⊢ ∃𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
43	15, 42	bnj1131 32667	1 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ ciun 4921  dom cdm 5580  suc csuc 6253   Fn wfn 6413  ‘cfv 6418  ωcom 7687   predc-bnj14 32567   trClc-bnj18 32573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-dm 5590  df-iota 6376  df-fv 6426  df-bnj18 32574

This theorem is referenced by:  bnj1015  32842