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Theorem dfrtrcl3 37503
Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 13736. (Contributed by RP, 5-Jun-2020.)
Assertion
Ref Expression
dfrtrcl3 t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
Distinct variable group:   𝑛,𝑟

Proof of Theorem dfrtrcl3
Dummy variables 𝑘 𝑎 𝑡 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rtrcl 13661 . 2 t* = (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
2 relexp0g 13696 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3 nn0ex 11242 . . . . . . . . 9 0 ∈ V
4 0nn0 11251 . . . . . . . . 9 0 ∈ ℕ0
5 oveq1 6611 . . . . . . . . . . . . 13 (𝑎 = 𝑡 → (𝑎𝑟𝑛) = (𝑡𝑟𝑛))
65iuneq2d 4513 . . . . . . . . . . . 12 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑡𝑟𝑛))
7 oveq2 6612 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑡𝑟𝑛) = (𝑡𝑟𝑘))
87cbviunv 4525 . . . . . . . . . . . 12 𝑛 ∈ ℕ0 (𝑡𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘)
96, 8syl6eq 2671 . . . . . . . . . . 11 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
109cbvmptv 4710 . . . . . . . . . 10 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑡 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
1110ov2ssiunov2 37470 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
123, 4, 11mp3an23 1413 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
132, 12eqsstr3d 3619 . . . . . . 7 (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
14 relexp1g 13700 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) = 𝑟)
15 1nn0 11252 . . . . . . . . 9 1 ∈ ℕ0
1610ov2ssiunov2 37470 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
173, 15, 16mp3an23 1413 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
1814, 17eqsstr3d 3619 . . . . . . 7 (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
19 nn0uz 11666 . . . . . . . 8 0 = (ℤ‘0)
2010iunrelexpuztr 37489 . . . . . . . 8 ((𝑟 ∈ V ∧ ℕ0 = (ℤ‘0) ∧ 0 ∈ ℕ0) → (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
2119, 4, 20mp3an23 1413 . . . . . . 7 (𝑟 ∈ V → (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
22 fvex 6158 . . . . . . . 8 ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ V
23 sseq2 3606 . . . . . . . . . . 11 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))
24 sseq2 3606 . . . . . . . . . . 11 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → (𝑟𝑧𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))
25 id 22 . . . . . . . . . . . . 13 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → 𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
2625, 25coeq12d 5246 . . . . . . . . . . . 12 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → (𝑧𝑧) = (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))
2726, 25sseq12d 3613 . . . . . . . . . . 11 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → ((𝑧𝑧) ⊆ 𝑧 ↔ (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))
2823, 24, 273anbi123d 1396 . . . . . . . . . 10 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))))
2928a1i 11 . . . . . . . . 9 (𝑟 ∈ V → (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))))
3029alrimiv 1852 . . . . . . . 8 (𝑟 ∈ V → ∀𝑧(𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))))
31 elabgt 3330 . . . . . . . 8 ((((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ V ∧ ∀𝑧(𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))))) → (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))))
3222, 30, 31sylancr 694 . . . . . . 7 (𝑟 ∈ V → (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))))
3313, 18, 21, 32mpbir3and 1243 . . . . . 6 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
34 intss1 4457 . . . . . 6 (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
3533, 34syl 17 . . . . 5 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
36 vex 3189 . . . . . . . . 9 𝑠 ∈ V
37 sseq2 3606 . . . . . . . . . 10 (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38 sseq2 3606 . . . . . . . . . 10 (𝑧 = 𝑠 → (𝑟𝑧𝑟𝑠))
39 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑠𝑧 = 𝑠)
4039, 39coeq12d 5246 . . . . . . . . . . 11 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
4140, 39sseq12d 3613 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
4237, 38, 413anbi123d 1396 . . . . . . . . 9 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
4336, 42elab 3333 . . . . . . . 8 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
44 eqid 2621 . . . . . . . . . 10 0 = ℕ0
4510iunrelexpmin2 37482 . . . . . . . . . 10 ((𝑟 ∈ V ∧ ℕ0 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4644, 45mpan2 706 . . . . . . . . 9 (𝑟 ∈ V → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
474619.21bi 2057 . . . . . . . 8 (𝑟 ∈ V → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4843, 47syl5bi 232 . . . . . . 7 (𝑟 ∈ V → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4948ralrimiv 2959 . . . . . 6 (𝑟 ∈ V → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠)
50 ssint 4458 . . . . . 6 (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠)
5149, 50sylibr 224 . . . . 5 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
5235, 51eqssd 3600 . . . 4 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
53 oveq1 6611 . . . . . 6 (𝑎 = 𝑟 → (𝑎𝑟𝑛) = (𝑟𝑟𝑛))
5453iuneq2d 4513 . . . . 5 (𝑎 = 𝑟 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
55 eqid 2621 . . . . 5 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))
56 ovex 6632 . . . . . 6 (𝑟𝑟𝑛) ∈ V
573, 56iunex 7093 . . . . 5 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) ∈ V
5854, 55, 57fvmpt 6239 . . . 4 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
5952, 58eqtrd 2655 . . 3 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
6059mpteq2ia 4700 . 2 (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
611, 60eqtri 2643 1 t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2907  Vcvv 3186  cun 3553  wss 3555   cint 4440   ciun 4485  cmpt 4673   I cid 4984  dom cdm 5074  ran crn 5075  cres 5076  ccom 5078  cfv 5847  (class class class)co 6604  0cc0 9880  1c1 9881  0cn0 11236  cuz 11631  t*crtcl 13659  𝑟crelexp 13694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-seq 12742  df-rtrcl 13661  df-relexp 13695
This theorem is referenced by:  brfvrtrcld  37504  fvrtrcllb0d  37505  fvrtrcllb0da  37506  fvrtrcllb1d  37507  dfrtrcl4  37508  corcltrcl  37509  cotrclrcl  37512
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