Theorem bnj893 35063

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 35061	. 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 3443	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
7		fveq1 6832	. . . . . . . . . . . 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 3781	. . . . . . . . . 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 6832	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓‘suc 𝑖) = (𝑔‘suc 𝑖))
12		fveq1 6832	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑖) = (𝑔‘𝑖))
13	12	iuneq1d 4973	. . . . . . . . . . . . . 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 3158	. . . . . . . . . . 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 3781	. . . . . . . . . 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 35059	. . . . . . . 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 2827	. . . . . . 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 625	. . . . . 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 3078	. . . . 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 2802	. . . 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 4947	. . . 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 4947	. . . 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 2768	. . 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 35060	. . . 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 3443	. . . . . . 7 ⊢ 𝑓 ∈ V
38	37	dmex 7851	. . . . . 6 ⊢ dom 𝑓 ∈ V
39		fvex 6846	. . . . . 6 ⊢ (𝑓‘𝑖) ∈ V
40	38, 39	iunex 7912	. . . . 5 ⊢ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ∈ V
41	40	rgenw 3054	. . . 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 7907	. . . 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 587	. . 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 2839	. 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 2839	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2713  ∀wral 3050  ∃wrex 3059  Vcvv 3439  [wsbc 3739   ∖ cdif 3897  ∅c0 4284  {csn 4579  ∪ ciun 4945  dom cdm 5623  suc csuc 6318   Fn wfn 6486  ‘cfv 6491  ωcom 7808   predc-bnj14 34823   FrSe w-bnj15 34827   trClc-bnj18 34829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-reg 9499  ax-inf2 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-om 7809  df-1o 8397  df-bnj17 34822  df-bnj14 34824  df-bnj13 34826  df-bnj15 34828  df-bnj18 34830

This theorem is referenced by:  bnj1125  35127  bnj1136  35132  bnj1177  35141  bnj1413  35170  bnj1452  35187