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Theorem tcrank 9820
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 9729 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦))
2 onsuc 7746 . . . . 5 (𝑦 ∈ On → suc 𝑦 ∈ On)
3 fveq2 6842 . . . . . . . 8 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43raleqdv 3313 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
5 fveq2 6842 . . . . . . . . 9 (𝑧 = 𝑢 → (rank‘𝑧) = (rank‘𝑢))
6 fveq2 6842 . . . . . . . . . 10 (𝑧 = 𝑢 → (TC‘𝑧) = (TC‘𝑢))
76imaeq2d 6013 . . . . . . . . 9 (𝑧 = 𝑢 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝑢)))
85, 7sseq12d 3977 . . . . . . . 8 (𝑧 = 𝑢 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
98cbvralvw 3225 . . . . . . 7 (∀𝑧 ∈ (𝑅1𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
104, 9bitrdi 286 . . . . . 6 (𝑥 = 𝑦 → (∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
11 fveq2 6842 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
1211raleqdv 3313 . . . . . 6 (𝑥 = suc 𝑦 → (∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
13 simpr 485 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)))
14 simprl 769 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → 𝑧 ∈ (𝑅1𝑥))
15 simplr 767 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
16 rankr1ai 9734 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1𝑥) → (rank‘𝑧) ∈ 𝑥)
17 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑦 = (rank‘𝑧) → (𝑅1𝑦) = (𝑅1‘(rank‘𝑧)))
1817raleqdv 3313 . . . . . . . . . . . . . . . . 17 (𝑦 = (rank‘𝑧) → (∀𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ↔ ∀𝑢 ∈ (𝑅1‘(rank‘𝑧))(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
1918rspcv 3577 . . . . . . . . . . . . . . . 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 9732 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑅1𝑥) → 𝑧 (𝑅1 “ On))
22 r1rankidb 9740 . . . . . . . . . . . . . . . 16 (𝑧 (𝑅1 “ On) → 𝑧 ⊆ (𝑅1‘(rank‘𝑧)))
23 ssralv 4010 . . . . . . . . . . . . . . . 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 47 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑅1𝑥) → (∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑢𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢))))
2614, 15, 25sylc 65 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∀𝑢𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
27 rankval3b 9762 . . . . . . . . . . . . . . . . . . . 20 (𝑧 (𝑅1 “ On) → (rank‘𝑧) = {𝑥 ∈ On ∣ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥})
2827eleq2d 2823 . . . . . . . . . . . . . . . . . . 19 (𝑧 (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) ↔ 𝑤 {𝑥 ∈ On ∣ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥}))
2928biimpd 228 . . . . . . . . . . . . . . . . . 18 (𝑧 (𝑅1 “ On) → (𝑤 ∈ (rank‘𝑧) → 𝑤 {𝑥 ∈ On ∣ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥}))
30 rankon 9731 . . . . . . . . . . . . . . . . . . . 20 (rank‘𝑧) ∈ On
3130oneli 6431 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ On)
32 eleq2w 2821 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((rank‘𝑢) ∈ 𝑥 ↔ (rank‘𝑢) ∈ 𝑤))
3332ralbidv 3174 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → (∀𝑢𝑧 (rank‘𝑢) ∈ 𝑥 ↔ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤))
3433onnminsb 7734 . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ¬ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤)
39 rexnal 3103 . . . . . . . . . . . . . . 15 (∃𝑢𝑧 ¬ (rank‘𝑢) ∈ 𝑤 ↔ ¬ ∀𝑢𝑧 (rank‘𝑢) ∈ 𝑤)
4038, 39sylibr 233 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → ∃𝑢𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
4140adantl 482 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑢𝑧 ¬ (rank‘𝑢) ∈ 𝑤)
42 r19.29 3117 . . . . . . . . . . . . 13 ((∀𝑢𝑧 (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ∃𝑢𝑧 ¬ (rank‘𝑢) ∈ 𝑤) → ∃𝑢𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤))
4326, 41, 42syl2anc 584 . . . . . . . . . . . 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 1137 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑢𝑧)
45 tcid 9675 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → 𝑧 ⊆ (TC‘𝑧))
4645elv 3451 . . . . . . . . . . . . . . . 16 𝑧 ⊆ (TC‘𝑧)
4746sseli 3940 . . . . . . . . . . . . . . 15 (𝑢𝑧𝑢 ∈ (TC‘𝑧))
48 fveqeq2 6851 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → ((rank‘𝑥) = 𝑤 ↔ (rank‘𝑢) = 𝑤))
4948rspcev 3581 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (TC‘𝑧) ∧ (rank‘𝑢) = 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
5049ex 413 . . . . . . . . . . . . . . 15 (𝑢 ∈ (TC‘𝑧) → ((rank‘𝑢) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
5144, 47, 503syl 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))
52 simp3l 1201 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (rank‘𝑢) ⊆ (rank “ (TC‘𝑢)))
5352sseld 3943 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank‘𝑢) → 𝑤 ∈ (rank “ (TC‘𝑢))))
54 simp1l 1197 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → 𝑧 ∈ (𝑅1𝑥))
55 rankf 9730 . . . . . . . . . . . . . . . . . . 19 rank: (𝑅1 “ On)⟶On
56 ffn 6668 . . . . . . . . . . . . . . . . . . 19 (rank: (𝑅1 “ On)⟶On → rank Fn (𝑅1 “ On))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . . 18 rank Fn (𝑅1 “ On)
58 r1tr 9712 . . . . . . . . . . . . . . . . . . . 20 Tr (𝑅1𝑥)
59 trel 5231 . . . . . . . . . . . . . . . . . . . 20 (Tr (𝑅1𝑥) → ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → 𝑢 ∈ (𝑅1𝑥)))
6058, 59ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → 𝑢 ∈ (𝑅1𝑥))
61 r1elwf 9732 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝑅1𝑥) → 𝑢 (𝑅1 “ On))
62 tcwf 9819 . . . . . . . . . . . . . . . . . . . 20 (𝑢 (𝑅1 “ On) → (TC‘𝑢) ∈ (𝑅1 “ On))
63 fvex 6855 . . . . . . . . . . . . . . . . . . . . 21 (TC‘𝑢) ∈ V
6463r1elss 9742 . . . . . . . . . . . . . . . . . . . 20 ((TC‘𝑢) ∈ (𝑅1 “ On) ↔ (TC‘𝑢) ⊆ (𝑅1 “ On))
6562, 64sylib 217 . . . . . . . . . . . . . . . . . . 19 (𝑢 (𝑅1 “ On) → (TC‘𝑢) ⊆ (𝑅1 “ On))
6660, 61, 653syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → (TC‘𝑢) ⊆ (𝑅1 “ On))
67 fvelimab 6914 . . . . . . . . . . . . . . . . . 18 ((rank Fn (𝑅1 “ On) ∧ (TC‘𝑢) ⊆ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
6857, 66, 67sylancr 587 . . . . . . . . . . . . . . . . 17 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) ↔ ∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤))
69 vex 3449 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
7069tcel 9681 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑧 → (TC‘𝑢) ⊆ (TC‘𝑧))
71 ssrexv 4011 . . . . . . . . . . . . . . . . . . 19 ((TC‘𝑢) ⊆ (TC‘𝑧) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7270, 71syl 17 . . . . . . . . . . . . . . . . . 18 (𝑢𝑧 → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7372adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → (∃𝑥 ∈ (TC‘𝑢)(rank‘𝑥) = 𝑤 → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7468, 73sylbid 239 . . . . . . . . . . . . . . . 16 ((𝑢𝑧𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑢)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
7544, 54, 74syl2anc 584 . . . . . . . . . . . . . . 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))
7653, 75syld 47 . . . . . . . . . . . . . 14 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → (𝑤 ∈ (rank‘𝑢) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
77 rankon 9731 . . . . . . . . . . . . . . . . . . 19 (rank‘𝑢) ∈ On
78 eloni 6327 . . . . . . . . . . . . . . . . . . . 20 ((rank‘𝑢) ∈ On → Ord (rank‘𝑢))
79 eloni 6327 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ On → Ord 𝑤)
80 ordtri3or 6349 . . . . . . . . . . . . . . . . . . . 20 ((Ord (rank‘𝑢) ∧ Ord 𝑤) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
8178, 79, 80syl2an 596 . . . . . . . . . . . . . . . . . . 19 (((rank‘𝑢) ∈ On ∧ 𝑤 ∈ On) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
8277, 31, 81sylancr 587 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
83 3orass 1090 . . . . . . . . . . . . . . . . . 18 (((rank‘𝑢) ∈ 𝑤 ∨ (rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)) ↔ ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢))))
8482, 83sylib 217 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (rank‘𝑧) → ((rank‘𝑢) ∈ 𝑤 ∨ ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢))))
8584orcanai 1001 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ (rank‘𝑧) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
8685ad2ant2l 744 . . . . . . . . . . . . . . 15 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
87863adant2 1131 . . . . . . . . . . . . . 14 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ((rank‘𝑢) = 𝑤𝑤 ∈ (rank‘𝑢)))
8851, 76, 87mpjaod 858 . . . . . . . . . . . . 13 (((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) ∧ 𝑢𝑧 ∧ ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤)) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
8988rexlimdv3a 3156 . . . . . . . . . . . 12 ((𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧)) → (∃𝑢𝑧 ((rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) ∧ ¬ (rank‘𝑢) ∈ 𝑤) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9013, 43, 89sylc 65 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ (𝑧 ∈ (𝑅1𝑥) ∧ 𝑤 ∈ (rank‘𝑧))) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤)
9190expr 457 . . . . . . . . . 10 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank‘𝑧) → ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
92 tcwf 9819 . . . . . . . . . . . . 13 (𝑧 (𝑅1 “ On) → (TC‘𝑧) ∈ (𝑅1 “ On))
93 r1elssi 9741 . . . . . . . . . . . . . 14 ((TC‘𝑧) ∈ (𝑅1 “ On) → (TC‘𝑧) ⊆ (𝑅1 “ On))
94 fvelimab 6914 . . . . . . . . . . . . . 14 ((rank Fn (𝑅1 “ On) ∧ (TC‘𝑧) ⊆ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9593, 94sylan2 593 . . . . . . . . . . . . 13 ((rank Fn (𝑅1 “ On) ∧ (TC‘𝑧) ∈ (𝑅1 “ On)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9657, 92, 95sylancr 587 . . . . . . . . . . . 12 (𝑧 (𝑅1 “ On) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9721, 96syl 17 . . . . . . . . . . 11 (𝑧 ∈ (𝑅1𝑥) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9897adantl 482 . . . . . . . . . 10 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank “ (TC‘𝑧)) ↔ ∃𝑥 ∈ (TC‘𝑧)(rank‘𝑥) = 𝑤))
9991, 98sylibrd 258 . . . . . . . . 9 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1𝑥)) → (𝑤 ∈ (rank‘𝑧) → 𝑤 ∈ (rank “ (TC‘𝑧))))
10099ssrdv 3950 . . . . . . . 8 (((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) ∧ 𝑧 ∈ (𝑅1𝑥)) → (rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
101100ralrimiva 3143 . . . . . . 7 ((𝑥 ∈ On ∧ ∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢))) → ∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
102101ex 413 . . . . . 6 (𝑥 ∈ On → (∀𝑦𝑥𝑢 ∈ (𝑅1𝑦)(rank‘𝑢) ⊆ (rank “ (TC‘𝑢)) → ∀𝑧 ∈ (𝑅1𝑥)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧))))
10310, 12, 102tfis3 7794 . . . . 5 (suc 𝑦 ∈ On → ∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)))
104 fveq2 6842 . . . . . . 7 (𝑧 = 𝐴 → (rank‘𝑧) = (rank‘𝐴))
105 fveq2 6842 . . . . . . . 8 (𝑧 = 𝐴 → (TC‘𝑧) = (TC‘𝐴))
106105imaeq2d 6013 . . . . . . 7 (𝑧 = 𝐴 → (rank “ (TC‘𝑧)) = (rank “ (TC‘𝐴)))
107104, 106sseq12d 3977 . . . . . 6 (𝑧 = 𝐴 → ((rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) ↔ (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
108107rspccv 3578 . . . . 5 (∀𝑧 ∈ (𝑅1‘suc 𝑦)(rank‘𝑧) ⊆ (rank “ (TC‘𝑧)) → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
1092, 103, 1083syl 18 . . . 4 (𝑦 ∈ On → (𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴))))
110109rexlimiv 3145 . . 3 (∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑦) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
1111, 110sylbi 216 . 2 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank “ (TC‘𝐴)))
112 tcvalg 9674 . . . 4 (𝐴 (𝑅1 “ On) → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
113 r1rankidb 9740 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
114 r1tr 9712 . . . . 5 Tr (𝑅1‘(rank‘𝐴))
115 fvex 6855 . . . . . . 7 (𝑅1‘(rank‘𝐴)) ∈ V
116 sseq2 3970 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (𝐴𝑥𝐴 ⊆ (𝑅1‘(rank‘𝐴))))
117 treq 5230 . . . . . . . 8 (𝑥 = (𝑅1‘(rank‘𝐴)) → (Tr 𝑥 ↔ Tr (𝑅1‘(rank‘𝐴))))
118116, 117anbi12d 631 . . . . . . 7 (𝑥 = (𝑅1‘(rank‘𝐴)) → ((𝐴𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴)))))
119115, 118elab 3630 . . . . . 6 ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))))
120 intss1 4924 . . . . . 6 ((𝑅1‘(rank‘𝐴)) ∈ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
121119, 120sylbir 234 . . . . 5 ((𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))) → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
122113, 114, 121sylancl 586 . . . 4 (𝐴 (𝑅1 “ On) → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ (𝑅1‘(rank‘𝐴)))
123112, 122eqsstrd 3982 . . 3 (𝐴 (𝑅1 “ On) → (TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)))
124 imass2 6054 . . . 4 ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank “ (𝑅1‘(rank‘𝐴))))
125 ffun 6671 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → Fun rank)
12655, 125ax-mp 5 . . . . . . 7 Fun rank
127 fvelima 6908 . . . . . . 7 ((Fun rank ∧ 𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴)))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
128126, 127mpan 688 . . . . . 6 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → ∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥)
129 rankr1ai 9734 . . . . . . . 8 (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → (rank‘𝑦) ∈ (rank‘𝐴))
130 eleq1 2825 . . . . . . . 8 ((rank‘𝑦) = 𝑥 → ((rank‘𝑦) ∈ (rank‘𝐴) ↔ 𝑥 ∈ (rank‘𝐴)))
131129, 130syl5ibcom 244 . . . . . . 7 (𝑦 ∈ (𝑅1‘(rank‘𝐴)) → ((rank‘𝑦) = 𝑥𝑥 ∈ (rank‘𝐴)))
132131rexlimiv 3145 . . . . . 6 (∃𝑦 ∈ (𝑅1‘(rank‘𝐴))(rank‘𝑦) = 𝑥𝑥 ∈ (rank‘𝐴))
133128, 132syl 17 . . . . 5 (𝑥 ∈ (rank “ (𝑅1‘(rank‘𝐴))) → 𝑥 ∈ (rank‘𝐴))
134133ssriv 3948 . . . 4 (rank “ (𝑅1‘(rank‘𝐴))) ⊆ (rank‘𝐴)
135124, 134sstrdi 3956 . . 3 ((TC‘𝐴) ⊆ (𝑅1‘(rank‘𝐴)) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
136123, 135syl 17 . 2 (𝐴 (𝑅1 “ On) → (rank “ (TC‘𝐴)) ⊆ (rank‘𝐴))
137111, 136eqssd 3961 1 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910   cuni 4865   cint 4907  Tr wtr 5222  cima 5636  Ord word 6316  Oncon0 6317  suc csuc 6319  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  TCctc 9672  𝑅1cr1 9698  rankcrnk 9699
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-tc 9673  df-r1 9700  df-rank 9701
This theorem is referenced by:  hsmexlem5  10366  grur1  10756
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