Theorem bnj1014 35258

Description: Technical lemma for bnj69 35307. 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 2926	. . . . . . . . 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 35229	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	5	nf5i 2182	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
7	2, 6	nfrexw 3312	. . . . . . . 8 ⊢ Ⅎ𝑖∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
8	7	nfab 2932	. . . . . . 7 ⊢ Ⅎ𝑖{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
9	1, 8	nfcxfr 2924	. . . . . 6 ⊢ Ⅎ𝑖𝐵
10	9	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑖 𝑔 ∈ 𝐵
11		nfv 1936	. . . . 5 ⊢ Ⅎ𝑖 𝑗 ∈ dom 𝑔
12	10, 11	nfan 1921	. . . 4 ⊢ Ⅎ𝑖(𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔)
13		nfv 1936	. . . 4 ⊢ Ⅎ𝑖(𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)
14	12, 13	nfim 1918	. . 3 ⊢ Ⅎ𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
15	14	nf5ri 2232	. 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 2847	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ dom 𝑔 ↔ 𝑖 ∈ dom 𝑔))
17	16	anbi2d 639	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) ↔ (𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)))
18		fveq2 6869	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑔‘𝑗) = (𝑔‘𝑖))
19	18	sseq1d 3969	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
20	17, 19	imbi12d 346	. . . 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 2042	. . 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 35118	. . . . . . 7 ⊢ (𝑔 ∈ 𝐵 → ∀𝑓 𝑔 ∈ 𝐵)
23	22	nf5i 2182	. . . . . 6 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐵
24		nfv 1936	. . . . . 6 ⊢ Ⅎ𝑓 𝑖 ∈ dom 𝑔
25	23, 24	nfan 1921	. . . . 5 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)
26		nfv 1936	. . . . 5 ⊢ Ⅎ𝑓(𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)
27	25, 26	nfim 1918	. . . 4 ⊢ Ⅎ𝑓((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
28		eleq1w 2847	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐵 ↔ 𝑔 ∈ 𝐵))
29		dmeq 5881	. . . . . . 7 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
30	29	eleq2d 2850	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ dom 𝑔))
31	28, 30	anbi12d 641	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)))
32		fveq1 6868	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑖) = (𝑔‘𝑖))
33	32	sseq1d 3969	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
34	31, 33	imbi12d 346	. . . 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 5007	. . . . 5 ⊢ (𝑖 ∈ dom 𝑓 → (𝑓‘𝑖) ⊆ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
36		ssiun2 5007	. . . . . 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 35223	. . . . . 6 ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
39	36, 38	sseqtrrdi 3979	. . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
40	35, 39	sylan9ssr 3952	. . . 4 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
41	27, 34, 40	chvarfv 2277	. . 3 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
42	21, 41	speivw 1995	. 2 ⊢ ∃𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
43	15, 42	bnj1131 35085	1 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∃wrex 3088   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287  {csn 4584  ∪ ciun 4951  dom cdm 5649  suc csuc 6350   Fn wfn 6518  ‘cfv 6523  ωcom 7848   predc-bnj14 34986   trClc-bnj18 34992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-dm 5659  df-iota 6479  df-fv 6531  df-bnj18 34993

This theorem is referenced by:  bnj1015  35259