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Theorem tcrank 8630
 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.
StepHypRef Expression
1 rankwflemb 8539 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦))
2 suceloni 6905 . . . . 5 (𝑦 ∈ On → suc 𝑦 ∈ On)
3 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43raleqdv 3121 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
5 fveq2 6103 . . . . . . . . 9 (𝑧 = 𝑢 → (rank‘𝑧) = (rank‘𝑢))
6 fveq2 6103 . . . . . . . . . 10 (𝑧 = 𝑢 → (TC‘𝑧) = (TC‘𝑢))
76imaeq2d 5385 . . . . . . . . 9 (𝑧 = 𝑢 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝑢)))
85, 7sseq12d 3597 . . . . . . . 8 (𝑧 = 𝑢 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
98cbvralv 3147 . . . . . . 7 (∀𝑧 ∈ (𝑅1𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
104, 9syl6bb 275 . . . . . 6 (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
11 fveq2 6103 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
1211raleqdv 3121 . . . . . 6 (𝑥 = suc 𝑦 → (∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
13 simpr 476 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)))
14 simprl 790 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → 𝑧 ∈ (𝑅1𝑥))
15 simplr 788 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
16 rankr1ai 8544 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1𝑥) → (rank‘𝑧) ∈ 𝑥)
17 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑦 = (rank‘𝑧) → (𝑅1𝑦) = (𝑅1‘(rank‘𝑧)))
1817raleqdv 3121 . . . . . . . . . . . . . . . . 17 (𝑦 = (rank‘𝑧) → (∀𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ↔ ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
1918rspcv 3278 . . . . . . . . . . . . . . . 16 ((rank‘𝑧) ∈ 𝑥 → (∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
2016, 19syl 17 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝑅1𝑥) → (∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
21 r1elwf 8542 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1𝑥) → 𝑧 (𝑅1 “ On))
22 r1rankidb 8550 . . . . . . . . . . . . . . . 16 (𝑧 (𝑅1 “ On) → 𝑧 ⊆ (𝑅1‘(rank‘𝑧)))
23 ssralv 3629 . . . . . . . . . . . . . . . 16 (𝑧 ⊆ (𝑅1‘(rank‘𝑧)) → (∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
2421, 22, 233syl 18 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝑅1𝑥) → (∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
2520, 24syld 46 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑅1𝑥) → (∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
2614, 15, 25sylc 63 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∀𝑢𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
27 rankval3b 8572 . . . . . . . . . . . . . . . . . . . 20 (𝑧 (𝑅1 “ On) → (rank‘𝑧) = {𝑥 ∈ On ∣ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥})
2827eleq2d 2673 . . . . . . . . . . . . . . . . . . 19 (𝑧 (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) ↔ 𝑤 {𝑥 ∈ On ∣ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥}))
2928biimpd 218 . . . . . . . . . . . . . . . . . 18 (𝑧 (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → 𝑤 {𝑥 ∈ On ∣ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥}))
30 rankon 8541 . . . . . . . . . . . . . . . . . . . 20 (rank‘𝑧) ∈ On
3130oneli 5752 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ On)
32 eleq2 2677 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((rank‘𝑢) ∈ 𝑥 ↔ (rank‘𝑢) ∈ 𝑤))
3332ralbidv 2969 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → (∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥 ↔ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤))
3433onnminsb 6896 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ On → (𝑤 {𝑥 ∈ On ∣ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥} → ¬ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤))
3531, 34syl 17 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (rank‘𝑧) → (𝑤 {𝑥 ∈ On ∣ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥} → ¬ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤))
3629, 35sylcom 30 . . . . . . . . . . . . . . . . 17 (𝑧 (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → ¬ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤))
3721, 36syl 17 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1𝑥) → (𝑤 ∈ (rank‘𝑧) → ¬ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤))
3837imp 444 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ¬ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤)
39 rexnal 2978 . . . . . . . . . . . . . . 15 (∃𝑢𝑧 ¬ (rank‘𝑢) ∈ 𝑤 ↔ ¬ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤)
4038, 39sylibr 223 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ∃𝑢𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
4140adantl 481 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑢𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
42 r19.29 3054 . . . . . . . . . . . . 13 ((∀𝑢𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ∃𝑢𝑧 ¬ (rank‘𝑢) ∈ 𝑤) → ∃𝑢𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤))
4326, 41, 42syl2anc 691 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑢𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤))
44 simp2 1055 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑢𝑧)
45 vex 3176 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
46 tcid 8498 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → 𝑧 ⊆ (TC‘𝑧))
4745, 46ax-mp 5 . . . . . . . . . . . . . . . 16 𝑧 ⊆ (TC‘𝑧)
4847sseli 3564 . . . . . . . . . . . . . . 15 (𝑢𝑧𝑢 ∈ (TC‘𝑧))
49 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑢 → (rank‘𝑥) = (rank‘𝑢))
5049eqeq1d 2612 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → ((rank‘𝑥) = 𝑤 ↔ (rank‘𝑢) = 𝑤))
5150rspcev 3282 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (TC‘𝑧) ∧ (rank‘𝑢) = 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
5251ex 449 . . . . . . . . . . . . . . 15 (𝑢 ∈ (TC‘𝑧) → ((rank‘𝑢) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
5344, 48, 523syl 18 . . . . . . . . . . . . . 14 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
54 simp3l 1082 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
5554sseld 3567 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank‘𝑢) → 𝑤 ∈ (rank “ (TC‘𝑢))))
56 simp1l 1078 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑧 ∈ (𝑅1𝑥))
57 rankf 8540 . . . . . . . . . . . . . . . . . . 19 rank: (𝑅1 “ On)⟶On
58 ffn 5958 . . . . . . . . . . . . . . . . . . 19 (rank: (𝑅1 “ On)⟶On → rank Fn (𝑅1 “ On))
5957, 58ax-mp 5 . . . . . . . . . . . . . . . . . 18 rank Fn (𝑅1 “ On)
60 r1tr 8522 . . . . . . . . . . . . . . . . . . . 20 Tr (𝑅1𝑥)
61 trel 4687 . . . . . . . . . . . . . . . . . . . 20 (Tr (𝑅1𝑥) → ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → 𝑢 ∈ (𝑅1𝑥)))
6260, 61ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → 𝑢 ∈ (𝑅1𝑥))
63 r1elwf 8542 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝑅1𝑥) → 𝑢 (𝑅1 “ On))
64 tcwf 8629 . . . . . . . . . . . . . . . . . . . 20 (𝑢 (𝑅1 “ On) → (TC‘𝑢) ∈ (𝑅1 “ On))
65 fvex 6113 . . . . . . . . . . . . . . . . . . . . 21 (TC‘𝑢) ∈ V
6665r1elss 8552 . . . . . . . . . . . . . . . . . . . 20 ((TC‘𝑢) ∈ (𝑅1 “ On) ↔ (TC‘𝑢) ⊆ (𝑅1 “ On))
6764, 66sylib 207 . . . . . . . . . . . . . . . . . . 19 (𝑢 (𝑅1 “ On) → (TC‘𝑢) ⊆ (𝑅1 “ On))
6862, 63, 673syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → (TC‘𝑢) ⊆ (𝑅1 “ On))
69 fvelimab 6163 . . . . . . . . . . . . . . . . . 18 ((rank Fn (𝑅1 “ On) ∧ (TC‘𝑢) ⊆ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
7059, 68, 69sylancr 694 . . . . . . . . . . . . . . . . 17 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
7145tcel 8504 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑧 → (TC‘𝑢) ⊆ (TC‘𝑧))
72 ssrexv 3630 . . . . . . . . . . . . . . . . . . 19 ((TC‘𝑢) ⊆ (TC‘𝑧) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7371, 72syl 17 . . . . . . . . . . . . . . . . . 18 (𝑢𝑧 → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7473adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7570, 74sylbid 229 . . . . . . . . . . . . . . . 16 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7644, 56, 75syl2anc 691 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank “ (TC‘𝑢)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7755, 76syld 46 . . . . . . . . . . . . . 14 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank‘𝑢) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
78 rankon 8541 . . . . . . . . . . . . . . . . . . 19 (rank‘𝑢) ∈ On
79 eloni 5650 . . . . . . . . . . . . . . . . . . . 20 ((rank‘𝑢) ∈ On → Ord (rank‘𝑢))
80 eloni 5650 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ On → Ord 𝑤)
81 ordtri3or 5672 . . . . . . . . . . . . . . . . . . . 20 ((Ord (rank‘𝑢) ∧ Ord 𝑤) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
8279, 80, 81syl2an 493 . . . . . . . . . . . . . . . . . . 19 (((rank‘𝑢) ∈ On ∧ 𝑤 ∈ On) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
8378, 31, 82sylancr 694 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
84 3orass 1034 . . . . . . . . . . . . . . . . . 18 (((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)) ↔ ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢))))
8583, 84sylib 207 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢))))
8685orcanai 950 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (rank‘𝑧) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
8786ad2ant2l 778 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
88873adant2 1073 . . . . . . . . . . . . . 14 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
8953, 77, 88mpjaod 395 . . . . . . . . . . . . 13 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
9089rexlimdv3a 3015 . . . . . . . . . . . 12 ((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → (∃𝑢𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9113, 43, 90sylc 63 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
9291expr 641 . . . . . . . . . 10 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank‘𝑧) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
93 tcwf 8629 . . . . . . . . . . . . 13 (𝑧 (𝑅1 “ On) → (TC‘𝑧) ∈ (𝑅1 “ On))
94 r1elssi 8551 . . . . . . . . . . . . . 14 ((TC‘𝑧) ∈ (𝑅1 “ On) → (TC‘𝑧) ⊆ (𝑅1 “ On))
95 fvelimab 6163 . . . . . . . . . . . . . 14 ((rank Fn (𝑅1 “ On) ∧ (TC‘𝑧) ⊆ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9694, 95sylan2 490 . . . . . . . . . . . . 13 ((rank Fn (𝑅1 “ On) ∧ (TC‘𝑧) ∈ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9759, 93, 96sylancr 694 . . . . . . . . . . . 12 (𝑧 (𝑅1 “ On) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9821, 97syl 17 . . . . . . . . . . 11 (𝑧 ∈ (𝑅1𝑥) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9998adantl 481 . . . . . . . . . 10 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
10092, 99sylibrd 248 . . . . . . . . 9 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ (rank “ (TC‘𝑧))))
101100ssrdv 3574 . . . . . . . 8 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1𝑥)) → (rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
102101ralrimiva 2949 . . . . . . 7 ((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) → ∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
103102ex 449 . . . . . 6 (𝑥 ∈ On → (∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
10410, 12, 103tfis3 6949 . . . . 5 (suc 𝑦 ∈ On → ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
105 fveq2 6103 . . . . . . 7 (𝑧 = 𝐴 → (rank‘𝑧) = (rank‘𝐴))
106 fveq2 6103 . . . . . . . 8 (𝑧 = 𝐴 → (TC‘𝑧) = (TC‘𝐴))
107106imaeq2d 5385 . . . . . . 7 (𝑧 = 𝐴 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝐴)))
108105, 107sseq12d 3597 . . . . . 6 (𝑧 = 𝐴 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
109108rspccv 3279 . . . . 5 (∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
1102, 104, 1093syl 18 . . . 4 (𝑦 ∈ On → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
111110rexlimiv 3009 . . 3 (∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
1121, 111sylbi 206 . 2 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
113 tcvalg 8497 . . . 4 (𝐴 (𝑅1 “ On) → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
114 r1rankidb 8550 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
115 r1tr 8522 . . . . 5 Tr (𝑅1‘(rank‘𝐴))
116 fvex 6113 . . . . . . 7 (𝑅1‘(rank‘𝐴)) ∈ V
117 sseq2 3590 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (𝐴𝑥𝐴 ⊆ (𝑅1‘(rank‘𝐴))))
118 treq 4686 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (Tr 𝑥 ↔ Tr (𝑅1‘(rank‘𝐴))))
119117, 118anbi12d 743 . . . . . . 7 (𝑥 = (𝑅1‘(rank‘𝐴)) → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴)))))
120116, 119elab 3319 . . . . . 6 ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))))
121 intss1 4427 . . . . . 6 ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
122120, 121sylbir 224 . . . . 5 ((𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))) → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
123114, 115, 122sylancl 693 . . . 4 (𝐴 (𝑅1 “ On) → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
124113, 123eqsstrd 3602 . . 3 (𝐴 (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)))
125 imass2 5420 . . . 4 ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank “ (𝑅1‘(rank‘𝐴))))
126 ffun 5961 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → Fun rank)
12757, 126ax-mp 5 . . . . . . 7 Fun rank
128 fvelima 6158 . . . . . . 7 ((Fun rank ∧ 𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴)))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
129127, 128mpan 702 . . . . . 6 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
130 rankr1ai 8544 . . . . . . . 8 (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → (rank‘𝑦) ∈ (rank‘𝐴))
131 eleq1 2676 . . . . . . . 8 ((rank‘𝑦) = 𝑥 → ((rank‘𝑦) ∈ (rank‘𝐴) ↔ 𝑥 ∈ (rank‘𝐴)))
132130, 131syl5ibcom 234 . . . . . . 7 (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → ((rank‘𝑦) = 𝑥𝑥 ∈ (rank‘𝐴)))
133132rexlimiv 3009 . . . . . 6 (∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥𝑥 ∈ (rank‘𝐴))
134129, 133syl 17 . . . . 5 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → 𝑥 ∈ (rank‘𝐴))
135134ssriv 3572 . . . 4 (rank “ (𝑅1‘(rank‘𝐴))) ⊆ (rank‘𝐴)
136125, 135syl6ss 3580 . . 3 ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
137124, 136syl 17 . 2 (𝐴 (𝑅1 “ On) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
138112, 137eqssd 3585 1 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372  ∩ cint 4410  Tr wtr 4680   “ cima 5041  Ord word 5639  Oncon0 5640  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  TCctc 8495  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-tc 8496  df-r1 8510  df-rank 8511 This theorem is referenced by:  hsmexlem5  9135  grur1  9521
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