Theorem bnj1145 30315

Description: Technical lemma for bnj69 30332. 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
bnj1145.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1145.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1145.3	⊢ 𝐷 = (ω ∖ {∅})
bnj1145.4	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj1145.5	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1145.6	⊢ (𝜃 ↔ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))

Assertion

Ref	Expression
bnj1145	⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴

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

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑛)   𝑋(𝑗)

Proof of Theorem bnj1145

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1145.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj1145.2	. . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj1145.3	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
4		bnj1145.4	. . 3 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
5	1, 2, 3, 4	bnj882 30250	. 2 ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
6		ss2iun 4472	. . . 4 ⊢ (∀𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ 𝐴 → ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ ∪ 𝑓 ∈ 𝐵 𝐴)
7		bnj1145.5	. . . . . . 7 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
8	7, 4	bnj1083 30300	. . . . . 6 ⊢ (𝑓 ∈ 𝐵 ↔ ∃𝑛𝜒)
9	2	bnj1095 30106	. . . . . . . . 9 ⊢ (𝜓 → ∀𝑖𝜓)
10	9, 7	bnj1096 30107	. . . . . . . 8 ⊢ (𝜒 → ∀𝑖𝜒)
11	3	bnj1098 30108	. . . . . . . . . . . . . . . . 17 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
12	7	bnj1232 30128	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
13	12	3anim3i 1243	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
14	11, 13	bnj1101 30109	. . . . . . . . . . . . . . . 16 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
15		ancl 567	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))))
16	14, 15	bnj101 30043	. . . . . . . . . . . . . . 15 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))
17		bnj1145.6	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 ↔ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))
18	17	imbi2i 325	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → 𝜃) ↔ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))))
19	18	exbii 1764	. . . . . . . . . . . . . . 15 ⊢ (∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → 𝜃) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))))
20	16, 19	mpbir 220	. . . . . . . . . . . . . 14 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → 𝜃)
21		bnj213 30206	. . . . . . . . . . . . . . . 16 ⊢ pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
22	21	bnj226 30056	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
23		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → 𝑖 = suc 𝑗)
24	17, 23	simplbiim 657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜃 → 𝑖 = suc 𝑗)
25		simp2 1055	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → 𝑖 ∈ 𝑛)
26	12	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → 𝑛 ∈ 𝐷)
27	3	bnj923 30092	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
28		elnn 6967	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑖 ∈ ω)
29	27, 28	sylan2 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → 𝑖 ∈ ω)
30	25, 26, 29	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → 𝑖 ∈ ω)
31	17, 30	bnj832 30082	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜃 → 𝑖 ∈ ω)
32		vex 3176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑗 ∈ V
33	32	bnj216 30054	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = suc 𝑗 → 𝑗 ∈ 𝑖)
34		elnn 6967	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑖 ∧ 𝑖 ∈ ω) → 𝑗 ∈ ω)
35	33, 34	sylan 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 = suc 𝑗 ∧ 𝑖 ∈ ω) → 𝑗 ∈ ω)
36	24, 31, 35	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 → 𝑗 ∈ ω)
37	17, 25	bnj832 30082	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜃 → 𝑖 ∈ 𝑛)
38	24, 37	eqeltrrd 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 → suc 𝑗 ∈ 𝑛)
39	2	bnj589 30233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜓 ↔ ∀𝑗 ∈ ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
40	39	biimpi 205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜓 → ∀𝑗 ∈ ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
41	40	bnj708 30080	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑗 ∈ ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
42		rsp 2913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑗 ∈ ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
43	41, 42	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
44	7, 43	sylbi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
45	44	3ad2ant3 1077	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
46	17, 45	bnj832 30082	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
47	36, 38, 46	mp2d 47	. . . . . . . . . . . . . . . 16 ⊢ (𝜃 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
48		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = suc 𝑗 → (𝑓‘𝑖) = (𝑓‘suc 𝑗))
49	48	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = suc 𝑗 → ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
50	24, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜃 → ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
51	47, 50	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ (𝜃 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
52	22, 51	bnj1262 30135	. . . . . . . . . . . . . 14 ⊢ (𝜃 → (𝑓‘𝑖) ⊆ 𝐴)
53	20, 52	bnj1023 30105	. . . . . . . . . . . . 13 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)
54		3anass 1035	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) ↔ (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜒)))
55	54	imbi1i 338	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) ↔ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴))
56	55	exbii 1764	. . . . . . . . . . . . 13 ⊢ (∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴))
57	53, 56	mpbi 219	. . . . . . . . . . . 12 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴)
58	1	biimpi 205	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
59	7, 58	bnj771 30088	. . . . . . . . . . . . . 14 ⊢ (𝜒 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
60		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = ∅ → (𝑓‘𝑖) = (𝑓‘∅))
61		bnj213 30206	. . . . . . . . . . . . . . . 16 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
62		sseq1 3589	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) → ((𝑓‘∅) ⊆ 𝐴 ↔ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴))
63	61, 62	mpbiri 247	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) → (𝑓‘∅) ⊆ 𝐴)
64		sseq1 3589	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑖) = (𝑓‘∅) → ((𝑓‘𝑖) ⊆ 𝐴 ↔ (𝑓‘∅) ⊆ 𝐴))
65	64	biimpar 501	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑖) = (𝑓‘∅) ∧ (𝑓‘∅) ⊆ 𝐴) → (𝑓‘𝑖) ⊆ 𝐴)
66	60, 63, 65	syl2an 493	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓‘𝑖) ⊆ 𝐴)
67	59, 66	sylan2 490	. . . . . . . . . . . . 13 ⊢ ((𝑖 = ∅ ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)
68	67	adantrl 748	. . . . . . . . . . . 12 ⊢ ((𝑖 = ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴)
69	57, 68	bnj1109 30111	. . . . . . . . . . 11 ⊢ ∃𝑗((𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)
70		19.9v 1883	. . . . . . . . . . 11 ⊢ (∃𝑗((𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) ↔ ((𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴))
71	69, 70	mpbi 219	. . . . . . . . . 10 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)
72	71	expcom 450	. . . . . . . . 9 ⊢ (𝜒 → (𝑖 ∈ 𝑛 → (𝑓‘𝑖) ⊆ 𝐴))
73		fndm 5904	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
74	7, 73	bnj770 30087	. . . . . . . . . 10 ⊢ (𝜒 → dom 𝑓 = 𝑛)
75		eleq2 2677	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑛 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑛))
76	75	imbi1d 330	. . . . . . . . . 10 ⊢ (dom 𝑓 = 𝑛 → ((𝑖 ∈ dom 𝑓 → (𝑓‘𝑖) ⊆ 𝐴) ↔ (𝑖 ∈ 𝑛 → (𝑓‘𝑖) ⊆ 𝐴)))
77	74, 76	syl 17	. . . . . . . . 9 ⊢ (𝜒 → ((𝑖 ∈ dom 𝑓 → (𝑓‘𝑖) ⊆ 𝐴) ↔ (𝑖 ∈ 𝑛 → (𝑓‘𝑖) ⊆ 𝐴)))
78	72, 77	mpbird 246	. . . . . . . 8 ⊢ (𝜒 → (𝑖 ∈ dom 𝑓 → (𝑓‘𝑖) ⊆ 𝐴))
79	10, 78	hbralrimi 2937	. . . . . . 7 ⊢ (𝜒 → ∀𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ 𝐴)
80	79	exlimiv 1845	. . . . . 6 ⊢ (∃𝑛𝜒 → ∀𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ 𝐴)
81	8, 80	sylbi 206	. . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∀𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ 𝐴)
82		ss2iun 4472	. . . . . 6 ⊢ (∀𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ 𝐴 → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ ∪ 𝑖 ∈ dom 𝑓 𝐴)
83		bnj1143 30115	. . . . . 6 ⊢ ∪ 𝑖 ∈ dom 𝑓 𝐴 ⊆ 𝐴
84	82, 83	syl6ss 3580	. . . . 5 ⊢ (∀𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ 𝐴 → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ 𝐴)
85	81, 84	syl 17	. . . 4 ⊢ (𝑓 ∈ 𝐵 → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ 𝐴)
86	6, 85	mprg 2910	. . 3 ⊢ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ ∪ 𝑓 ∈ 𝐵 𝐴
87	4	bnj1317 30146	. . . 4 ⊢ (𝑤 ∈ 𝐵 → ∀𝑓 𝑤 ∈ 𝐵)
88	87	bnj1146 30116	. . 3 ⊢ ∪ 𝑓 ∈ 𝐵 𝐴 ⊆ 𝐴
89	86, 88	sstri 3577	. 2 ⊢ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ 𝐴
90	5, 89	eqsstri 3598	1 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ ciun 4455  dom cdm 5038  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957   ∧ w-bnj17 30005   predc-bnj14 30007   trClc-bnj18 30013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fn 5807  df-fv 5812  df-om 6958  df-bnj17 30006  df-bnj14 30008  df-bnj18 30014

This theorem is referenced by:  bnj1147  30316