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Theorem dfrtrcl3 42484
Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 15009. (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 14935 . 2 t* = (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
2 relexp0g 14969 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3 nn0ex 12478 . . . . . . . . 9 0 ∈ V
4 0nn0 12487 . . . . . . . . 9 0 ∈ ℕ0
5 oveq1 7416 . . . . . . . . . . . . 13 (𝑎 = 𝑡 → (𝑎𝑟𝑛) = (𝑡𝑟𝑛))
65iuneq2d 5027 . . . . . . . . . . . 12 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑡𝑟𝑛))
7 oveq2 7417 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑡𝑟𝑛) = (𝑡𝑟𝑘))
87cbviunv 5044 . . . . . . . . . . . 12 𝑛 ∈ ℕ0 (𝑡𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘)
96, 8eqtrdi 2789 . . . . . . . . . . 11 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
109cbvmptv 5262 . . . . . . . . . 10 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑡 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
1110ov2ssiunov2 42451 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
123, 4, 11mp3an23 1454 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
132, 12eqsstrrd 4022 . . . . . . 7 (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
14 relexp1g 14973 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) = 𝑟)
15 1nn0 12488 . . . . . . . . 9 1 ∈ ℕ0
1610ov2ssiunov2 42451 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
173, 15, 16mp3an23 1454 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
1814, 17eqsstrrd 4022 . . . . . . 7 (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
19 nn0uz 12864 . . . . . . . 8 0 = (ℤ‘0)
2010iunrelexpuztr 42470 . . . . . . . 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 1454 . . . . . . 7 (𝑟 ∈ V → (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
22 fvex 6905 . . . . . . . 8 ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ V
23 sseq2 4009 . . . . . . . . . . 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 4009 . . . . . . . . . . 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 5865 . . . . . . . . . . . 12 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → (𝑧𝑧) = (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))
2726, 25sseq12d 4016 . . . . . . . . . . 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 1437 . . . . . . . . . 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 1931 . . . . . . . 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 3663 . . . . . . . 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 588 . . . . . . 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 1343 . . . . . 6 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
34 intss1 4968 . . . . . 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 3479 . . . . . . . . 9 𝑠 ∈ V
37 sseq2 4009 . . . . . . . . . 10 (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38 sseq2 4009 . . . . . . . . . 10 (𝑧 = 𝑠 → (𝑟𝑧𝑟𝑠))
39 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑠𝑧 = 𝑠)
4039, 39coeq12d 5865 . . . . . . . . . . 11 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
4140, 39sseq12d 4016 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
4237, 38, 413anbi123d 1437 . . . . . . . . 9 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
4336, 42elab 3669 . . . . . . . 8 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
44 eqid 2733 . . . . . . . . . 10 0 = ℕ0
4510iunrelexpmin2 42463 . . . . . . . . . 10 ((𝑟 ∈ V ∧ ℕ0 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4644, 45mpan2 690 . . . . . . . . 9 (𝑟 ∈ V → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
474619.21bi 2183 . . . . . . . 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 3146 . . . . . 6 (𝑟 ∈ V → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠)
50 ssint 4969 . . . . . 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 4000 . . . 4 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
53 oveq1 7416 . . . . . 6 (𝑎 = 𝑟 → (𝑎𝑟𝑛) = (𝑟𝑟𝑛))
5453iuneq2d 5027 . . . . 5 (𝑎 = 𝑟 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
55 eqid 2733 . . . . 5 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))
56 ovex 7442 . . . . . 6 (𝑟𝑟𝑛) ∈ V
573, 56iunex 7955 . . . . 5 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) ∈ V
5854, 55, 57fvmpt 6999 . . . 4 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
5952, 58eqtrd 2773 . . 3 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
6059mpteq2ia 5252 . 2 (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
611, 60eqtri 2761 1 t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1088  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wral 3062  Vcvv 3475  cun 3947  wss 3949   cint 4951   ciun 4998  cmpt 5232   I cid 5574  dom cdm 5677  ran crn 5678  cres 5679  ccom 5681  cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  0cn0 12472  cuz 12822  t*crtcl 14933  𝑟crelexp 14966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-seq 13967  df-rtrcl 14935  df-relexp 14967
This theorem is referenced by:  brfvrtrcld  42485  fvrtrcllb0d  42486  fvrtrcllb0da  42487  fvrtrcllb1d  42488  dfrtrcl4  42489  corcltrcl  42490  cotrclrcl  42493
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