Theorem bnj906 31542

Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj906	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))

Proof of Theorem bnj906

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

Step	Hyp	Ref	Expression
1		1onn 7991	. . . . . . . 8 ⊢ 1o ∈ ω
2		1n0 7847	. . . . . . . 8 ⊢ 1o ≠ ∅
3		eldifsn 4538	. . . . . . . 8 ⊢ (1o ∈ (ω ∖ {∅}) ↔ (1o ∈ ω ∧ 1o ≠ ∅))
4	1, 2, 3	mpbir2an 702	. . . . . . 7 ⊢ 1o ∈ (ω ∖ {∅})
5	4	ne0ii 4155	. . . . . 6 ⊢ (ω ∖ {∅}) ≠ ∅
6		biid 253	. . . . . . 7 ⊢ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
7		biid 253	. . . . . . 7 ⊢ (∀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)))
8		eqid 2825	. . . . . . 7 ⊢ (ω ∖ {∅}) = (ω ∖ {∅})
9	6, 7, 8	bnj852 31533	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ (ω ∖ {∅})∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
10		r19.2z 4284	. . . . . 6 ⊢ (((ω ∖ {∅}) ≠ ∅ ∧ ∀𝑛 ∈ (ω ∖ {∅})∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))) → ∃𝑛 ∈ (ω ∖ {∅})∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
11	5, 9, 10	sylancr 581	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑛 ∈ (ω ∖ {∅})∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
12		euex 2650	. . . . 5 ⊢ (∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) → ∃𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
13	11, 12	bnj31 31330	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑛 ∈ (ω ∖ {∅})∃𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
14		rexcom4 3442	. . . 4 ⊢ (∃𝑛 ∈ (ω ∖ {∅})∃𝑓(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))))
15	13, 14	sylib 210	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃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))))
16		abid 2813	. . 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)))} ↔ ∃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))))
17	15, 16	bnj1198 31408	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑓 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)))})
18		simp2 1171	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
19	18	reximi 3219	. . . . . 6 ⊢ (∃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))) → ∃𝑛 ∈ (ω ∖ {∅})(𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
20	16, 19	sylbi 209	. . . . 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)))} → ∃𝑛 ∈ (ω ∖ {∅})(𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
21		df-rex 3123	. . . . . 6 ⊢ (∃𝑛 ∈ (ω ∖ {∅})(𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ ∃𝑛(𝑛 ∈ (ω ∖ {∅}) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
22		19.41v 2048	. . . . . . 7 ⊢ (∃𝑛(𝑛 ∈ (ω ∖ {∅}) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ↔ (∃𝑛 𝑛 ∈ (ω ∖ {∅}) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
23	22	simprbi 492	. . . . . 6 ⊢ (∃𝑛(𝑛 ∈ (ω ∖ {∅}) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
24	21, 23	sylbi 209	. . . . 5 ⊢ (∃𝑛 ∈ (ω ∖ {∅})(𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
25	20, 24	syl 17	. . . 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)))} → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
26		eqid 2825	. . . . . . 7 ⊢ {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)))}
27	8, 26	bnj900 31541	. . . . . 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)))} → ∅ ∈ dom 𝑓)
28		fveq2 6437	. . . . . . 7 ⊢ (𝑖 = ∅ → (𝑓‘𝑖) = (𝑓‘∅))
29	28	ssiun2s 4786	. . . . . 6 ⊢ (∅ ∈ dom 𝑓 → (𝑓‘∅) ⊆ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
30	27, 29	syl 17	. . . . 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)))} → (𝑓‘∅) ⊆ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
31		ssiun2 4785	. . . . . 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)))} → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ ∪ 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))
32	6, 7, 8, 26	bnj882 31538	. . . . . 6 ⊢ 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)
33	31, 32	syl6sseqr 3877	. . . . 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)))} → ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
34	30, 33	sstrd 3837	. . . 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)))} → (𝑓‘∅) ⊆ trCl(𝑋, 𝐴, 𝑅))
35	25, 34	eqsstr3d 3865	. . 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)))} → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
36	35	exlimiv 2029	. 2 ⊢ (∃𝑓 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)))} → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
37	17, 36	syl 17	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∃!weu 2639  {cab 2811   ≠ wne 2999  ∀wral 3117  ∃wrex 3118   ∖ cdif 3795   ⊆ wss 3798  ∅c0 4146  {csn 4399  ∪ ciun 4742  dom cdm 5346  suc csuc 5969   Fn wfn 6122  ‘cfv 6127  ωcom 7331  1_oc1o 7824   predc-bnj14 31299   FrSe w-bnj15 31303   trClc-bnj18 31305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-reg 8773  ax-inf2 8822

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-om 7332  df-1o 7831  df-bnj17 31298  df-bnj14 31300  df-bnj13 31302  df-bnj15 31304  df-bnj18 31306

This theorem is referenced by:  bnj1137  31605  bnj1136  31607  bnj1175  31614  bnj1177  31616  bnj1413  31645  bnj1408  31646  bnj1417  31651  bnj1442  31659  bnj1452  31662