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Theorem dfrtrcl3 38527
Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14001. (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 13928 . 2 t* = (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
2 relexp0g 13961 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3 nn0ex 11490 . . . . . . . . 9 0 ∈ V
4 0nn0 11499 . . . . . . . . 9 0 ∈ ℕ0
5 oveq1 6820 . . . . . . . . . . . . 13 (𝑎 = 𝑡 → (𝑎𝑟𝑛) = (𝑡𝑟𝑛))
65iuneq2d 4699 . . . . . . . . . . . 12 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑡𝑟𝑛))
7 oveq2 6821 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑡𝑟𝑛) = (𝑡𝑟𝑘))
87cbviunv 4711 . . . . . . . . . . . 12 𝑛 ∈ ℕ0 (𝑡𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘)
96, 8syl6eq 2810 . . . . . . . . . . 11 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
109cbvmptv 4902 . . . . . . . . . 10 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑡 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
1110ov2ssiunov2 38494 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
123, 4, 11mp3an23 1565 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
132, 12eqsstr3d 3781 . . . . . . 7 (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
14 relexp1g 13965 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) = 𝑟)
15 1nn0 11500 . . . . . . . . 9 1 ∈ ℕ0
1610ov2ssiunov2 38494 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
173, 15, 16mp3an23 1565 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
1814, 17eqsstr3d 3781 . . . . . . 7 (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
19 nn0uz 11915 . . . . . . . 8 0 = (ℤ‘0)
2010iunrelexpuztr 38513 . . . . . . . 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 1565 . . . . . . 7 (𝑟 ∈ V → (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
22 fvex 6362 . . . . . . . 8 ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ V
23 sseq2 3768 . . . . . . . . . . 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 3768 . . . . . . . . . . 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 5442 . . . . . . . . . . . 12 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → (𝑧𝑧) = (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))
2726, 25sseq12d 3775 . . . . . . . . . . 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 1548 . . . . . . . . . 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 2004 . . . . . . . 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 3487 . . . . . . . 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 698 . . . . . . 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 1428 . . . . . 6 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
34 intss1 4644 . . . . . 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 3343 . . . . . . . . 9 𝑠 ∈ V
37 sseq2 3768 . . . . . . . . . 10 (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38 sseq2 3768 . . . . . . . . . 10 (𝑧 = 𝑠 → (𝑟𝑧𝑟𝑠))
39 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑠𝑧 = 𝑠)
4039, 39coeq12d 5442 . . . . . . . . . . 11 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
4140, 39sseq12d 3775 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
4237, 38, 413anbi123d 1548 . . . . . . . . 9 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
4336, 42elab 3490 . . . . . . . 8 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
44 eqid 2760 . . . . . . . . . 10 0 = ℕ0
4510iunrelexpmin2 38506 . . . . . . . . . 10 ((𝑟 ∈ V ∧ ℕ0 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4644, 45mpan2 709 . . . . . . . . 9 (𝑟 ∈ V → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
474619.21bi 2206 . . . . . . . 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 3103 . . . . . 6 (𝑟 ∈ V → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠)
50 ssint 4645 . . . . . 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 3761 . . . 4 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
53 oveq1 6820 . . . . . 6 (𝑎 = 𝑟 → (𝑎𝑟𝑛) = (𝑟𝑟𝑛))
5453iuneq2d 4699 . . . . 5 (𝑎 = 𝑟 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
55 eqid 2760 . . . . 5 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))
56 ovex 6841 . . . . . 6 (𝑟𝑟𝑛) ∈ V
573, 56iunex 7312 . . . . 5 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) ∈ V
5854, 55, 57fvmpt 6444 . . . 4 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
5952, 58eqtrd 2794 . . 3 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
6059mpteq2ia 4892 . 2 (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
611, 60eqtri 2782 1 t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072  wal 1630   = wceq 1632  wcel 2139  {cab 2746  wral 3050  Vcvv 3340  cun 3713  wss 3715   cint 4627   ciun 4672  cmpt 4881   I cid 5173  dom cdm 5266  ran crn 5267  cres 5268  ccom 5270  cfv 6049  (class class class)co 6813  0cc0 10128  1c1 10129  0cn0 11484  cuz 11879  t*crtcl 13926  𝑟crelexp 13959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-seq 12996  df-rtrcl 13928  df-relexp 13960
This theorem is referenced by:  brfvrtrcld  38528  fvrtrcllb0d  38529  fvrtrcllb0da  38530  fvrtrcllb1d  38531  dfrtrcl4  38532  corcltrcl  38533  cotrclrcl  38536
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