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Theorem dfrtrcl3 42786
Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 15013. (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 14939 . 2 t* = (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
2 relexp0g 14973 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3 nn0ex 12482 . . . . . . . . 9 0 ∈ V
4 0nn0 12491 . . . . . . . . 9 0 ∈ ℕ0
5 oveq1 7418 . . . . . . . . . . . . 13 (𝑎 = 𝑡 → (𝑎𝑟𝑛) = (𝑡𝑟𝑛))
65iuneq2d 5025 . . . . . . . . . . . 12 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑡𝑟𝑛))
7 oveq2 7419 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑡𝑟𝑛) = (𝑡𝑟𝑘))
87cbviunv 5042 . . . . . . . . . . . 12 𝑛 ∈ ℕ0 (𝑡𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘)
96, 8eqtrdi 2786 . . . . . . . . . . 11 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
109cbvmptv 5260 . . . . . . . . . 10 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑡 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
1110ov2ssiunov2 42753 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
123, 4, 11mp3an23 1451 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
132, 12eqsstrrd 4020 . . . . . . 7 (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
14 relexp1g 14977 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) = 𝑟)
15 1nn0 12492 . . . . . . . . 9 1 ∈ ℕ0
1610ov2ssiunov2 42753 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
173, 15, 16mp3an23 1451 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
1814, 17eqsstrrd 4020 . . . . . . 7 (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
19 nn0uz 12868 . . . . . . . 8 0 = (ℤ‘0)
2010iunrelexpuztr 42772 . . . . . . . 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 1451 . . . . . . 7 (𝑟 ∈ V → (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
22 fvex 6903 . . . . . . . 8 ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ V
23 sseq2 4007 . . . . . . . . . . 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 4007 . . . . . . . . . . 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 5863 . . . . . . . . . . . 12 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → (𝑧𝑧) = (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))
2726, 25sseq12d 4014 . . . . . . . . . . 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 1434 . . . . . . . . . 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 1928 . . . . . . . 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 3661 . . . . . . . 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 585 . . . . . . 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 1340 . . . . . 6 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
34 intss1 4966 . . . . . 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 3476 . . . . . . . . 9 𝑠 ∈ V
37 sseq2 4007 . . . . . . . . . 10 (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38 sseq2 4007 . . . . . . . . . 10 (𝑧 = 𝑠 → (𝑟𝑧𝑟𝑠))
39 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑠𝑧 = 𝑠)
4039, 39coeq12d 5863 . . . . . . . . . . 11 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
4140, 39sseq12d 4014 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
4237, 38, 413anbi123d 1434 . . . . . . . . 9 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
4336, 42elab 3667 . . . . . . . 8 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
44 eqid 2730 . . . . . . . . . 10 0 = ℕ0
4510iunrelexpmin2 42765 . . . . . . . . . 10 ((𝑟 ∈ V ∧ ℕ0 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4644, 45mpan2 687 . . . . . . . . 9 (𝑟 ∈ V → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
474619.21bi 2180 . . . . . . . 8 (𝑟 ∈ V → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4843, 47biimtrid 241 . . . . . . 7 (𝑟 ∈ V → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4948ralrimiv 3143 . . . . . 6 (𝑟 ∈ V → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠)
50 ssint 4967 . . . . . 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 233 . . . . 5 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
5235, 51eqssd 3998 . . . 4 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
53 oveq1 7418 . . . . . 6 (𝑎 = 𝑟 → (𝑎𝑟𝑛) = (𝑟𝑟𝑛))
5453iuneq2d 5025 . . . . 5 (𝑎 = 𝑟 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
55 eqid 2730 . . . . 5 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))
56 ovex 7444 . . . . . 6 (𝑟𝑟𝑛) ∈ V
573, 56iunex 7957 . . . . 5 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) ∈ V
5854, 55, 57fvmpt 6997 . . . 4 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
5952, 58eqtrd 2770 . . 3 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
6059mpteq2ia 5250 . 2 (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
611, 60eqtri 2758 1 t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1085  wal 1537   = wceq 1539  wcel 2104  {cab 2707  wral 3059  Vcvv 3472  cun 3945  wss 3947   cint 4949   ciun 4996  cmpt 5230   I cid 5572  dom cdm 5675  ran crn 5676  cres 5677  ccom 5679  cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113  0cn0 12476  cuz 12826  t*crtcl 14937  𝑟crelexp 14970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13971  df-rtrcl 14939  df-relexp 14971
This theorem is referenced by:  brfvrtrcld  42787  fvrtrcllb0d  42788  fvrtrcllb0da  42789  fvrtrcllb1d  42790  dfrtrcl4  42791  corcltrcl  42792  cotrclrcl  42795
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