Theorem tcrank 9856

Description: This theorem expresses two different facts from the two subset implications in this equality. In the forward direction, it says that the transitive closure has members of every rank below 𝐴. Stated another way, to construct a set at a given rank, you have to climb the entire hierarchy of ordinals below (rank‘𝐴), constructing at least one set at each level in order to move up the ranks. In the reverse direction, it says that every member of (TC‘𝐴) has a rank below the rank of 𝐴, since intuitively it contains only the members of 𝐴 and the members of those and so on, but nothing "bigger" than 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)

Assertion

Ref	Expression
tcrank	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))

Proof of Theorem tcrank

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rankwflemb 9765	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦))
2		onsuc 7809	. . . . 5 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
3		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
4	3	raleqdv 3329	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
5		fveq2 6882	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (rank‘𝑧) = (rank‘𝑢))
6		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → (TC‘𝑧) = (TC‘𝑢))
7	6	imaeq2d 6063	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝑢)))
8	5, 7	sseq12d 3978	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
9	8	cbvralvw 3249	. . . . . . 7 ⊢ (∀𝑧 ∈ (𝑅1‘𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
10	4, 9	bitrdi 290	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
11		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
12	11	raleqdv 3329	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
13		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)))
14		simprl 782	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → 𝑧 ∈ (𝑅1‘𝑥))
15		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
16		rankr1ai 9770	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (rank‘𝑧) ∈ 𝑥)
17		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (rank‘𝑧) → (𝑅1‘𝑦) = (𝑅1‘(rank‘𝑧)))
18	17	raleqdv 3329	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (rank‘𝑧) → (∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ↔ ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
19	18	rspcv 3586	. . . . . . . . . . . . . . . 16 ⊢ ((rank‘𝑧) ∈ 𝑥 → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
20	16, 19	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
21		r1elwf 9768	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → 𝑧 ∈ ∪ (𝑅1 “ On))
22		r1rankidb 9776	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → 𝑧 ⊆ (𝑅1‘(rank‘𝑧)))
23		ssralv 4014	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ (𝑅1‘(rank‘𝑧)) → (∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
24	21, 22, 23	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
25	20, 24	syld 48	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
26	14, 15, 25	sylc 66	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
27		rankval3b 9798	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (rank‘𝑧) = ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥})
28	27	eleq2d 2855	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) ↔ 𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥}))
29	28	biimpd 232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥}))
30		rankon 9767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (rank‘𝑧) ∈ On
31	30	oneli 6477	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ On)
32		eleq2w 2853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑤 → ((rank‘𝑢) ∈ 𝑥 ↔ (rank‘𝑢) ∈ 𝑤))
33	32	ralbidv 3194	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑤 → (∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥 ↔ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
34	33	onnminsb 7798	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ On → (𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥} → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
35	31, 34	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (rank‘𝑧) → (𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥} → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
36	29, 35	sylcom 31	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
37	21, 36	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (𝑤 ∈ (rank‘𝑧) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
38	37	imp 411	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤)
39		rexnal 3123	. . . . . . . . . . . . . . 15 ⊢ (∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤 ↔ ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤)
40	38, 39	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
41	40	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
42		r19.29 3134	. . . . . . . . . . . . 13 ⊢ ((∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤) → ∃𝑢 ∈ 𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤))
43	26, 41, 42	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑢 ∈ 𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤))
44		simp2 1153	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑢 ∈ 𝑧)
45		tcid 9706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ V → 𝑧 ⊆ (TC‘𝑧))
46	45	elv 3468	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ⊆ (TC‘𝑧)
47	46	sseli 3941	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑧 → 𝑢 ∈ (TC‘𝑧))
48		fveqeq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑢 → ((rank‘𝑥) = 𝑤 ↔ (rank‘𝑢) = 𝑤))
49	48	rspcev 3590	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (TC‘𝑧) ∧ (rank‘𝑢) = 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
50	49	ex 417	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (TC‘𝑧) → ((rank‘𝑢) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
51	44, 47, 50	3syl 19	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
52		simp3l 1218	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
53	52	sseld 3944	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank‘𝑢) → 𝑤 ∈ (rank “ (TC‘𝑢))))
54		simp1l 1214	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑧 ∈ (𝑅1‘𝑥))
55		rankf 9766	. . . . . . . . . . . . . . . . . . 19 ⊢ rank:∪ (𝑅1 “ On)⟶On
56		ffn 6706	. . . . . . . . . . . . . . . . . . 19 ⊢ (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
57	55, 56	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ rank Fn ∪ (𝑅1 “ On)
58		r1tr 9748	. . . . . . . . . . . . . . . . . . . 20 ⊢ Tr (𝑅1‘𝑥)
59		trel 5230	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Tr (𝑅1‘𝑥) → ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → 𝑢 ∈ (𝑅1‘𝑥)))
60	58, 59	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → 𝑢 ∈ (𝑅1‘𝑥))
61		r1elwf 9768	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (𝑅1‘𝑥) → 𝑢 ∈ ∪ (𝑅1 “ On))
62		tcwf 9855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ ∪ (𝑅1 “ On) → (TC‘𝑢) ∈ ∪ (𝑅1 “ On))
63		fvex 6895	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TC‘𝑢) ∈ V
64	63	r1elss 9778	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((TC‘𝑢) ∈ ∪ (𝑅1 “ On) ↔ (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
65	62, 64	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ∪ (𝑅1 “ On) → (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
66	60, 61, 65	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
67		fvelimab 6954	. . . . . . . . . . . . . . . . . 18 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑢) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
68	57, 66, 67	sylancr 598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
69		vex 3467	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
70	69	tcel 9712	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ 𝑧 → (TC‘𝑢) ⊆ (TC‘𝑧))
71		ssrexv 4015	. . . . . . . . . . . . . . . . . . 19 ⊢ ((TC‘𝑢) ⊆ (TC‘𝑧) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
72	70, 71	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ 𝑧 → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
73	72	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
74	68, 73	sylbid 243	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
75	44, 54, 74	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank “ (TC‘𝑢)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
76	53, 75	syld 48	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank‘𝑢) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
77		rankon 9767	. . . . . . . . . . . . . . . . . . 19 ⊢ (rank‘𝑢) ∈ On
78		eloni 6371	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((rank‘𝑢) ∈ On → Ord (rank‘𝑢))
79		eloni 6371	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ On → Ord 𝑤)
80		ordtri3or 6394	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Ord (rank‘𝑢) ∧ Ord 𝑤) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
81	78, 79, 80	syl2an 607	. . . . . . . . . . . . . . . . . . 19 ⊢ (((rank‘𝑢) ∈ On ∧ 𝑤 ∈ On) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
82	77, 31, 81	sylancr 598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
83		3orass 1104	. . . . . . . . . . . . . . . . . 18 ⊢ (((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)) ↔ ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢))))
84	82, 83	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢))))
85	84	orcanai 1018	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ (rank‘𝑧) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
86	85	ad2ant2l 758	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
87	86	3adant2 1147	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
88	51, 76, 87	mpjaod 873	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
89	88	rexlimdv3a 3176	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → (∃𝑢 ∈ 𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
90	13, 43, 89	sylc 66	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
91	90	expr 461	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank‘𝑧) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
92		tcwf 9855	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ∈ ∪ (𝑅1 “ On))
93		r1elssi 9777	. . . . . . . . . . . . . 14 ⊢ ((TC‘𝑧) ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ⊆ ∪ (𝑅1 “ On))
94		fvelimab 6954	. . . . . . . . . . . . . 14 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
95	93, 94	sylan2 604	. . . . . . . . . . . . 13 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ∈ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
96	57, 92, 95	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
97	21, 96	syl 18	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
98	97	adantl 486	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
99	91, 98	sylibrd 262	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ (rank “ (TC‘𝑧))))
100	99	ssrdv 3951	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
101	100	ralrimiva 3163	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) → ∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
102	101	ex 417	. . . . . 6 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
103	10, 12, 102	tfis3 7854	. . . . 5 ⊢ (suc 𝑦 ∈ On → ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
104		fveq2 6882	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (rank‘𝑧) = (rank‘𝐴))
105		fveq2 6882	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (TC‘𝑧) = (TC‘𝐴))
106	105	imaeq2d 6063	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝐴)))
107	104, 106	sseq12d 3978	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
108	107	rspccv 3587	. . . . 5 ⊢ (∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
109	2, 103, 108	3syl 19	. . . 4 ⊢ (𝑦 ∈ On → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
110	109	rexlimiv 3165	. . 3 ⊢ (∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
111	1, 110	sylbi 220	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
112		tcvalg 9705	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
113		r1rankidb 9776	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
114		r1tr 9748	. . . . 5 ⊢ Tr (𝑅1‘(rank‘𝐴))
115		fvex 6895	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V
116		sseq2 3971	. . . . . . . 8 ⊢ (𝑥 = (𝑅1‘(rank‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))))
117		treq 5229	. . . . . . . 8 ⊢ (𝑥 = (𝑅1‘(rank‘𝐴)) → (Tr 𝑥 ↔ Tr (𝑅1‘(rank‘𝐴))))
118	116, 117	anbi12d 643	. . . . . . 7 ⊢ (𝑥 = (𝑅1‘(rank‘𝐴)) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴)))))
119	115, 118	elab 3647	. . . . . 6 ⊢ ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))))
120		intss1 4932	. . . . . 6 ⊢ ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
121	119, 120	sylbir 238	. . . . 5 ⊢ ((𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
122	113, 114, 121	sylancl 597	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
123	112, 122	eqsstrd 3979	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)))
124		imass2 6105	. . . 4 ⊢ ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank “ (𝑅1‘(rank‘𝐴))))
125		ffun 6709	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
126	55, 125	ax-mp 5	. . . . . . 7 ⊢ Fun rank
127		fvelima 6947	. . . . . . 7 ⊢ ((Fun rank ∧ 𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴)))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
128	126, 127	mpan 702	. . . . . 6 ⊢ (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
129		rankr1ai 9770	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → (rank‘𝑦) ∈ (rank‘𝐴))
130		eleq1 2857	. . . . . . . 8 ⊢ ((rank‘𝑦) = 𝑥 → ((rank‘𝑦) ∈ (rank‘𝐴) ↔ 𝑥 ∈ (rank‘𝐴)))
131	129, 130	syl5ibcom 248	. . . . . . 7 ⊢ (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → ((rank‘𝑦) = 𝑥 → 𝑥 ∈ (rank‘𝐴)))
132	131	rexlimiv 3165	. . . . . 6 ⊢ (∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥 → 𝑥 ∈ (rank‘𝐴))
133	128, 132	syl 18	. . . . 5 ⊢ (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → 𝑥 ∈ (rank‘𝐴))
134	133	ssriv 3949	. . . 4 ⊢ (rank “ (𝑅1‘(rank‘𝐴))) ⊆ (rank‘𝐴)
135	124, 134	sstrdi 3957	. . 3 ⊢ ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
136	123, 135	syl 18	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
137	111, 136	eqssd 3962	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1100   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ⊆ wss 3913  ∪ cuni 4876  ∩ cint 4916  Tr wtr 5222   “ cima 5665  Ord word 6360  Oncon0 6361  suc csuc 6363  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  TCctc 9703  𝑅₁cr1 9734  rankcrnk 9735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-tc 9704  df-r1 9736  df-rank 9737

This theorem is referenced by:  hsmexlem5  10414  grur1  10805