Theorem bnj893 32482

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 264	. . 3 ⊢ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		biid 264	. . 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 2739	. . 3 ⊢ (ω ∖ {∅}) = (ω ∖ {∅})
4		eqid 2739	. . 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 32480	. 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 3403	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
7		fveq1 6676	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
8	7	eqeq1d 2741	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅)))
9	6, 8	sbcie 3723	. . . . . . . . . 10 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅))
10	9	bicomi 227	. . . . . . . . 9 ⊢ ((𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
11		fveq1 6676	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓‘suc 𝑖) = (𝑔‘suc 𝑖))
12		fveq1 6676	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑖) = (𝑔‘𝑖))
13	12	iuneq1d 4909	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅))
14	11, 13	eqeq12d 2755	. . . . . . . . . . . . 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 344	. . . . . . . . . . . 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 3110	. . . . . . . . . . 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 3723	. . . . . . . . . 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 227	. . . . . . . . 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 32478	. . . . . . . 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 2825	. . . . . . 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 627	. . . . . 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 3160	. . . . 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 2804	. . . 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 4884	. . . 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 4884	. . . 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 2772	. . 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 264	. . . . 5 ⊢ ((𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑋, 𝐴, 𝑅))
28		biid 264	. . . . 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 2739	. . . . 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 264	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ (ω ∖ {∅})) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ (ω ∖ {∅})))
31		biid 264	. . . . 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 264	. . . . 5 ⊢ ([𝑧 / 𝑔](𝑔‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ [𝑧 / 𝑔](𝑔‘∅) = pred(𝑋, 𝐴, 𝑅))
33		biid 264	. . . . 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 264	. . . . 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 264	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
36	27, 28, 3, 29, 30, 31, 32, 33, 34, 35	bnj849 32479	. . . 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 3403	. . . . . . 7 ⊢ 𝑓 ∈ V
38	37	dmex 7645	. . . . . 6 ⊢ dom 𝑓 ∈ V
39		fvex 6690	. . . . . 6 ⊢ (𝑓‘𝑖) ∈ V
40	38, 39	iunex 7697	. . . . 5 ⊢ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ∈ V
41	40	rgenw 3066	. . . 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 7692	. . . 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 589	. . 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 2838	. 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 2838	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  {cab 2717  ∀wral 3054  ∃wrex 3055  Vcvv 3399  [wsbc 3681   ∖ cdif 3841  ∅c0 4212  {csn 4517  ∪ ciun 4882  dom cdm 5526  suc csuc 6175   Fn wfn 6335  ‘cfv 6340  ωcom 7602   predc-bnj14 32240   FrSe w-bnj15 32244   trClc-bnj18 32246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-reg 9132  ax-inf2 9180

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-om 7603  df-1o 8134  df-bnj17 32239  df-bnj14 32241  df-bnj13 32243  df-bnj15 32245  df-bnj18 32247

This theorem is referenced by:  bnj1125  32546  bnj1136  32551  bnj1177  32560  bnj1413  32589  bnj1452  32606