Theorem bnj893 34921

Description: Property of trCl. Under certain conditions, the transitive closure of 𝑋 in 𝐴 by 𝑅 is a set. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj893	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)

Proof of Theorem bnj893

Dummy variables 𝑓 𝑔 𝑖 𝑛 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		biid 261	. . 3 ⊢ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		biid 261	. . 3 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		eqid 2735	. . 3 ⊢ (ω ∖ {∅}) = (ω ∖ {∅})
4		eqid 2735	. . 3 ⊢ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))} = {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}
5	1, 2, 3, 4	bnj882 34919	. 2 ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
6		vex 3482	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
7		fveq1 6906	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
8	7	eqeq1d 2737	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅)))
9	6, 8	sbcie 3835	. . . . . . . . . 10 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅))
10	9	bicomi 224	. . . . . . . . 9 ⊢ ((𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
11		fveq1 6906	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓‘suc 𝑖) = (𝑔‘suc 𝑖))
12		fveq1 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑖) = (𝑔‘𝑖))
13	12	iuneq1d 5024	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))
14	11, 13	eqeq12d 2751	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))
15	14	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
16	15	ralbidv 3176	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
17	6, 16	sbcie 3835	. . . . . . . . . 10 ⊢ ([𝑔 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))
18	17	bicomi 224	. . . . . . . . 9 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝑔 / 𝑓]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
19	4, 10, 18	bnj873 34917	. . . . . . . 8 ⊢ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))} = {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}
20	19	eleq2i 2831	. . . . . . 7 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))} ↔ 𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))})
21	20	anbi1i 624	. . . . . 6 ⊢ ((𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))} ∧ 𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)) ↔ (𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))} ∧ 𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)))
22	21	rexbii2 3088	. . . . 5 ⊢ (∃𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
23	22	abbii 2807	. . . 4 ⊢ {𝑤 ∣ ∃𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)} = {𝑤 ∣ ∃𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)}
24		df-iun 4998	. . . 4 ⊢ ∪ 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) = {𝑤 ∣ ∃𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)}
25		df-iun 4998	. . . 4 ⊢ ∪ 𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) = {𝑤 ∣ ∃𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}𝑤 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)}
26	23, 24, 25	3eqtr4i 2773	. . 3 ⊢ ∪ 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) = ∪ 𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
27		biid 261	. . . . 5 ⊢ ((𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅))
28		biid 261	. . . . 5 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))
29		eqid 2735	. . . . 5 ⊢ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))} = {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}
30		biid 261	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ (ω ∖ {∅})) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ (ω ∖ {∅})))
31		biid 261	. . . . 5 ⊢ ((𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
32		biid 261	. . . . 5 ⊢ ([𝑧 / 𝑔](𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ [𝑧 / 𝑔](𝑔‘∅) = pred(𝑋, 𝐴, 𝑅))
33		biid 261	. . . . 5 ⊢ ([𝑧 / 𝑔]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝑧 / 𝑔]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))
34		biid 261	. . . . 5 ⊢ ([𝑧 / 𝑔](𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [𝑧 / 𝑔](𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))))
35		biid 261	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
36	27, 28, 3, 29, 30, 31, 32, 33, 34, 35	bnj849 34918	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))} ∈ V)
37		vex 3482	. . . . . . 7 ⊢ 𝑓 ∈ V
38	37	dmex 7932	. . . . . 6 ⊢ dom 𝑓 ∈ V
39		fvex 6920	. . . . . 6 ⊢ (𝑓‘𝑖) ∈ V
40	38, 39	iunex 7992	. . . . 5 ⊢ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ∈ V
41	40	rgenw 3063	. . . 4 ⊢ ∀𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ∈ V
42		iunexg 7987	. . . 4 ⊢ (({𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))} ∈ V ∧ ∀𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ∈ V) → ∪ 𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ∈ V)
43	36, 41, 42	sylancl 586	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ 𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑔 Fn 𝑛 ∧ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ∈ V)
44	26, 43	eqeltrid 2843	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ∈ V)
45	5, 44	eqeltrid 2843	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  ∃wrex 3068  Vcvv 3478  [wsbc 3791   ∖ cdif 3960  ∅c0 4339  {csn 4631  ∪ ciun 4996  dom cdm 5689  suc csuc 6388   Fn wfn 6558  ‘cfv 6563  ωcom 7887   predc-bnj14 34681   FrSe w-bnj15 34685   trClc-bnj18 34687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-bnj17 34680  df-bnj14 34682  df-bnj13 34684  df-bnj15 34686  df-bnj18 34688

This theorem is referenced by:  bnj1125  34985  bnj1136  34990  bnj1177  34999  bnj1413  35028  bnj1452  35045