Theorem tcrank 9307

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 9216	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦))
2		suceloni 7522	. . . . 5 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
3		fveq2 6664	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦))
4	3	raleqdv 3415	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
5		fveq2 6664	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (rank‘𝑧) = (rank‘𝑢))
6		fveq2 6664	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → (TC‘𝑧) = (TC‘𝑢))
7	6	imaeq2d 5923	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝑢)))
8	5, 7	sseq12d 3999	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
9	8	cbvralvw 3449	. . . . . . 7 ⊢ (∀𝑧 ∈ (𝑅1‘𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
10	4, 9	syl6bb 289	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
11		fveq2 6664	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦))
12	11	raleqdv 3415	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
13		simpr 487	. . . . . . . . . . . 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 769	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → 𝑧 ∈ (𝑅1‘𝑥))
15		simplr 767	. . . . . . . . . . . . . 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 9221	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (rank‘𝑧) ∈ 𝑥)
17		fveq2 6664	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (rank‘𝑧) → (𝑅1‘𝑦) = (𝑅1‘(rank‘𝑧)))
18	17	raleqdv 3415	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (rank‘𝑧) → (∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ↔ ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
19	18	rspcv 3617	. . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 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 9219	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → 𝑧 ∈ ∪ (𝑅1 “ On))
22		r1rankidb 9227	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → 𝑧 ⊆ (𝑅1‘(rank‘𝑧)))
23		ssralv 4032	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ 𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
25	20, 24	syld 47	. . . . . . . . . . . . . 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 65	. . . . . . . . . . . . 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 9249	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (rank‘𝑧) = ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥})
28	27	eleq2d 2898	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) ↔ 𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥}))
29	28	biimpd 231	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥}))
30		rankon 9218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (rank‘𝑧) ∈ On
31	30	oneli 6292	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ On)
32		eleq2w 2896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑤 → ((rank‘𝑢) ∈ 𝑥 ↔ (rank‘𝑢) ∈ 𝑤))
33	32	ralbidv 3197	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑤 → (∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥 ↔ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
34	33	onnminsb 7513	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ On → (𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥} → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
35	31, 34	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (rank‘𝑧) → (𝑤 ∈ ∩ {𝑥 ∈ On ∣ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑥} → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
36	29, 35	sylcom 30	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
37	21, 36	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (𝑤 ∈ (rank‘𝑧) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤))
38	37	imp 409	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤)
39		rexnal 3238	. . . . . . . . . . . . . . 15 ⊢ (∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤 ↔ ¬ ∀𝑢 ∈ 𝑧 (rank‘𝑢) ∈ 𝑤)
40	38, 39	sylibr 236	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ∃𝑢 ∈ 𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
41	40	adantl 484	. . . . . . . . . . . . 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 3254	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑢 ∈ 𝑧)
45		tcid 9175	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ V → 𝑧 ⊆ (TC‘𝑧))
46	45	elv 3499	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ⊆ (TC‘𝑧)
47	46	sseli 3962	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑧 → 𝑢 ∈ (TC‘𝑧))
48		fveqeq2 6673	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑢 → ((rank‘𝑥) = 𝑤 ↔ (rank‘𝑢) = 𝑤))
49	48	rspcev 3622	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (TC‘𝑧) ∧ (rank‘𝑢) = 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
50	49	ex 415	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (TC‘𝑧) → ((rank‘𝑢) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
51	44, 47, 50	3syl 18	. . . . . . . . . . . . . 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 1197	. . . . . . . . . . . . . . . 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 3965	. . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢 ∈ 𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑧 ∈ (𝑅1‘𝑥))
55		rankf 9217	. . . . . . . . . . . . . . . . . . 19 ⊢ rank:∪ (𝑅1 “ On)⟶On
56		ffn 6508	. . . . . . . . . . . . . . . . . . 19 ⊢ (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
57	55, 56	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ rank Fn ∪ (𝑅1 “ On)
58		r1tr 9199	. . . . . . . . . . . . . . . . . . . 20 ⊢ Tr (𝑅1‘𝑥)
59		trel 5171	. . . . . . . . . . . . . . . . . . . 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 9219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (𝑅1‘𝑥) → 𝑢 ∈ ∪ (𝑅1 “ On))
62		tcwf 9306	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ ∪ (𝑅1 “ On) → (TC‘𝑢) ∈ ∪ (𝑅1 “ On))
63		fvex 6677	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TC‘𝑢) ∈ V
64	63	r1elss 9229	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((TC‘𝑢) ∈ ∪ (𝑅1 “ On) ↔ (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
65	62, 64	sylib 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ∪ (𝑅1 “ On) → (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
66	60, 61, 65	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (TC‘𝑢) ⊆ ∪ (𝑅1 “ On))
67		fvelimab 6731	. . . . . . . . . . . . . . . . . 18 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑢) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
68	57, 66, 67	sylancr 589	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
69		vex 3497	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
70	69	tcel 9181	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ 𝑧 → (TC‘𝑢) ⊆ (TC‘𝑧))
71		ssrexv 4033	. . . . . . . . . . . . . . . . . . 19 ⊢ ((TC‘𝑢) ⊆ (TC‘𝑧) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
72	70, 71	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ 𝑧 → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
73	72	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
74	68, 73	sylbid 242	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ 𝑧 ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
75	44, 54, 74	syl2anc 586	. . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . 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 9218	. . . . . . . . . . . . . . . . . . 19 ⊢ (rank‘𝑢) ∈ On
78		eloni 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((rank‘𝑢) ∈ On → Ord (rank‘𝑢))
79		eloni 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ On → Ord 𝑤)
80		ordtri3or 6217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Ord (rank‘𝑢) ∧ Ord 𝑤) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
81	78, 79, 80	syl2an 597	. . . . . . . . . . . . . . . . . . 19 ⊢ (((rank‘𝑢) ∈ On ∧ 𝑤 ∈ On) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
82	77, 31, 81	sylancr 589	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
83		3orass 1086	. . . . . . . . . . . . . . . . . 18 ⊢ (((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)) ↔ ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢))))
84	82, 83	sylib 220	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢))))
85	84	orcanai 999	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ (rank‘𝑧) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
86	85	ad2ant2l 744	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ (𝑅1‘𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 ∨ 𝑤 ∈ (rank‘𝑢)))
87	86	3adant2 1127	. . . . . . . . . . . . . 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 856	. . . . . . . . . . . . 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 3286	. . . . . . . . . . . 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 65	. . . . . . . . . . 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 459	. . . . . . . . . 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 9306	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ∈ ∪ (𝑅1 “ On))
93		r1elssi 9228	. . . . . . . . . . . . . 14 ⊢ ((TC‘𝑧) ∈ ∪ (𝑅1 “ On) → (TC‘𝑧) ⊆ ∪ (𝑅1 “ On))
94		fvelimab 6731	. . . . . . . . . . . . . 14 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ⊆ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
95	93, 94	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ (TC‘𝑧) ∈ ∪ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
96	57, 92, 95	sylancr 589	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
97	21, 96	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑅1‘𝑥) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
98	97	adantl 484	. . . . . . . . . 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 261	. . . . . . . . 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 3972	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1‘𝑥)) → (rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
101	100	ralrimiva 3182	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ∀𝑢 ∈ (𝑅1‘𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) → ∀𝑧 ∈ (𝑅1‘𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
102	101	ex 415	. . . . . 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 7566	. . . . 5 ⊢ (suc 𝑦 ∈ On → ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
104		fveq2 6664	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (rank‘𝑧) = (rank‘𝐴))
105		fveq2 6664	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (TC‘𝑧) = (TC‘𝐴))
106	105	imaeq2d 5923	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝐴)))
107	104, 106	sseq12d 3999	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
108	107	rspccv 3619	. . . . 5 ⊢ (∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
109	2, 103, 108	3syl 18	. . . 4 ⊢ (𝑦 ∈ On → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
110	109	rexlimiv 3280	. . 3 ⊢ (∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
111	1, 110	sylbi 219	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
112		tcvalg 9174	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
113		r1rankidb 9227	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
114		r1tr 9199	. . . . 5 ⊢ Tr (𝑅1‘(rank‘𝐴))
115		fvex 6677	. . . . . . 7 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V
116		sseq2 3992	. . . . . . . 8 ⊢ (𝑥 = (𝑅1‘(rank‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))))
117		treq 5170	. . . . . . . 8 ⊢ (𝑥 = (𝑅1‘(rank‘𝐴)) → (Tr 𝑥 ↔ Tr (𝑅1‘(rank‘𝐴))))
118	116, 117	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = (𝑅1‘(rank‘𝐴)) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴)))))
119	115, 118	elab 3666	. . . . . 6 ⊢ ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))))
120		intss1 4883	. . . . . 6 ⊢ ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
121	119, 120	sylbir 237	. . . . 5 ⊢ ((𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
122	113, 114, 121	sylancl 588	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
123	112, 122	eqsstrd 4004	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)))
124		imass2 5959	. . . 4 ⊢ ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank “ (𝑅1‘(rank‘𝐴))))
125		ffun 6511	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
126	55, 125	ax-mp 5	. . . . . . 7 ⊢ Fun rank
127		fvelima 6725	. . . . . . 7 ⊢ ((Fun rank ∧ 𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴)))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
128	126, 127	mpan 688	. . . . . 6 ⊢ (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
129		rankr1ai 9221	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → (rank‘𝑦) ∈ (rank‘𝐴))
130		eleq1 2900	. . . . . . . 8 ⊢ ((rank‘𝑦) = 𝑥 → ((rank‘𝑦) ∈ (rank‘𝐴) ↔ 𝑥 ∈ (rank‘𝐴)))
131	129, 130	syl5ibcom 247	. . . . . . 7 ⊢ (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → ((rank‘𝑦) = 𝑥 → 𝑥 ∈ (rank‘𝐴)))
132	131	rexlimiv 3280	. . . . . 6 ⊢ (∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥 → 𝑥 ∈ (rank‘𝐴))
133	128, 132	syl 17	. . . . 5 ⊢ (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → 𝑥 ∈ (rank‘𝐴))
134	133	ssriv 3970	. . . 4 ⊢ (rank “ (𝑅1‘(rank‘𝐴))) ⊆ (rank‘𝐴)
135	124, 134	sstrdi 3978	. . 3 ⊢ ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
136	123, 135	syl 17	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
137	111, 136	eqssd 3983	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ⊆ wss 3935  ∪ cuni 4831  ∩ cint 4868  Tr wtr 5164   “ cima 5552  Ord word 6184  Oncon0 6185  suc csuc 6187  Fun wfun 6343   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  TCctc 9172  𝑅₁cr1 9185  rankcrnk 9186

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-tc 9173  df-r1 9187  df-rank 9188

This theorem is referenced by:  hsmexlem5  9846  grur1  10236