Theorem bnj1014 35658

Description: Technical lemma for bnj69 35707. 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 35629	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	5	nf5i 2186	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
7	2, 6	nfrexw 3314	. . . . . . . 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 1949	. . . . 5 ⊢ Ⅎ𝑖 𝑗 ∈ dom 𝑔
12	10, 11	nfan 1934	. . . 4 ⊢ Ⅎ𝑖(𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔)
13		nfv 1949	. . . 4 ⊢ Ⅎ𝑖(𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)
14	12, 13	nfim 1931	. . 3 ⊢ Ⅎ𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
15	14	nf5ri 2235	. 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 642	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) ↔ (𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)))
18		fveq2 6905	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑔‘𝑗) = (𝑔‘𝑖))
19	18	sseq1d 3944	. . . . 5 ⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
20	17, 19	imbi12d 347	. . . 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 2055	. . 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 35518	. . . . . . 7 ⊢ (𝑔 ∈ 𝐵 → ∀𝑓 𝑔 ∈ 𝐵)
23	22	nf5i 2186	. . . . . 6 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐵
24		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑓 𝑖 ∈ dom 𝑔
25	23, 24	nfan 1934	. . . . 5 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)
26		nfv 1949	. . . . 5 ⊢ Ⅎ𝑓(𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)
27	25, 26	nfim 1931	. . . 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 5891	. . . . . . 7 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
30	29	eleq2d 2850	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ dom 𝑔))
31	28, 30	anbi12d 644	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔)))
32		fveq1 6904	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑖) = (𝑔‘𝑖))
33	32	sseq1d 3944	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)))
34	31, 33	imbi12d 347	. . . 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 5009	. . . . 5 ⊢ (𝑖 ∈ dom 𝑓 → (𝑓‘𝑖) ⊆ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
36		ssiun2 5009	. . . . . 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 35623	. . . . . 6 ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
39	36, 38	sseqtrrdi 3954	. . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
40	35, 39	sylan9ssr 3927	. . . 4 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
41	27, 34, 40	chvarfv 2280	. . 3 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑖 ∈ dom 𝑔) → (𝑔‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
42	21, 41	speivw 2008	. 2 ⊢ ∃𝑖((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
43	15, 42	bnj1131 35485	1 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  {csn 4588  ∪ ciun 4954  dom cdm 5655  suc csuc 6379   Fn wfn 6550  ‘cfv 6555  ωcom 7898   predc-bnj14 35386   trClc-bnj18 35392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-dm 5665  df-iota 6509  df-fv 6563  df-bnj18 35393

This theorem is used by:  bnj1015  35659