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Theorem dfrtrcl3 38724
Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14101. (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 14028 . 2 t* = (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
2 relexp0g 14061 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3 nn0ex 11549 . . . . . . . . 9 0 ∈ V
4 0nn0 11559 . . . . . . . . 9 0 ∈ ℕ0
5 oveq1 6853 . . . . . . . . . . . . 13 (𝑎 = 𝑡 → (𝑎𝑟𝑛) = (𝑡𝑟𝑛))
65iuneq2d 4705 . . . . . . . . . . . 12 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑡𝑟𝑛))
7 oveq2 6854 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑡𝑟𝑛) = (𝑡𝑟𝑘))
87cbviunv 4717 . . . . . . . . . . . 12 𝑛 ∈ ℕ0 (𝑡𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘)
96, 8syl6eq 2815 . . . . . . . . . . 11 (𝑎 = 𝑡 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
109cbvmptv 4911 . . . . . . . . . 10 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑡 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑡𝑟𝑘))
1110ov2ssiunov2 38691 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
123, 4, 11mp3an23 1577 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟0) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
132, 12eqsstr3d 3802 . . . . . . 7 (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
14 relexp1g 14065 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) = 𝑟)
15 1nn0 11560 . . . . . . . . 9 1 ∈ ℕ0
1610ov2ssiunov2 38691 . . . . . . . . 9 ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
173, 15, 16mp3an23 1577 . . . . . . . 8 (𝑟 ∈ V → (𝑟𝑟1) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
1814, 17eqsstr3d 3802 . . . . . . 7 (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
19 nn0uz 11927 . . . . . . . 8 0 = (ℤ‘0)
2010iunrelexpuztr 38710 . . . . . . . 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 1577 . . . . . . 7 (𝑟 ∈ V → (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
22 fvex 6392 . . . . . . . 8 ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ V
23 sseq2 3789 . . . . . . . . . . 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 3789 . . . . . . . . . . 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 5457 . . . . . . . . . . . 12 (𝑧 = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) → (𝑧𝑧) = (((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟)))
2726, 25sseq12d 3796 . . . . . . . . . . 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 1560 . . . . . . . . . 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 2022 . . . . . . . 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 3502 . . . . . . . 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 581 . . . . . . 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 1442 . . . . . 6 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
34 intss1 4650 . . . . . 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 3353 . . . . . . . . 9 𝑠 ∈ V
37 sseq2 3789 . . . . . . . . . 10 (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38 sseq2 3789 . . . . . . . . . 10 (𝑧 = 𝑠 → (𝑟𝑧𝑟𝑠))
39 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑠𝑧 = 𝑠)
4039, 39coeq12d 5457 . . . . . . . . . . 11 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
4140, 39sseq12d 3796 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
4237, 38, 413anbi123d 1560 . . . . . . . . 9 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
4336, 42elab 3507 . . . . . . . 8 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
44 eqid 2765 . . . . . . . . . 10 0 = ℕ0
4510iunrelexpmin2 38703 . . . . . . . . . 10 ((𝑟 ∈ V ∧ ℕ0 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4644, 45mpan2 682 . . . . . . . . 9 (𝑟 ∈ V → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
474619.21bi 2221 . . . . . . . 8 (𝑟 ∈ V → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠𝑟𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4843, 47syl5bi 233 . . . . . . 7 (𝑟 ∈ V → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠))
4948ralrimiv 3112 . . . . . 6 (𝑟 ∈ V → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ 𝑠)
50 ssint 4651 . . . . . 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 225 . . . . 5 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
5235, 51eqssd 3780 . . . 4 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟))
53 oveq1 6853 . . . . . 6 (𝑎 = 𝑟 → (𝑎𝑟𝑛) = (𝑟𝑟𝑛))
5453iuneq2d 4705 . . . . 5 (𝑎 = 𝑟 𝑛 ∈ ℕ0 (𝑎𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
55 eqid 2765 . . . . 5 (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛)) = (𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))
56 ovex 6878 . . . . . 6 (𝑟𝑟𝑛) ∈ V
573, 56iunex 7349 . . . . 5 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) ∈ V
5854, 55, 57fvmpt 6475 . . . 4 (𝑟 ∈ V → ((𝑎 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑎𝑟𝑛))‘𝑟) = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
5952, 58eqtrd 2799 . . 3 (𝑟 ∈ V → {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
6059mpteq2ia 4901 . 2 (𝑟 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧𝑟𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
611, 60eqtri 2787 1 t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  w3a 1107  wal 1650   = wceq 1652  wcel 2155  {cab 2751  wral 3055  Vcvv 3350  cun 3732  wss 3734   cint 4635   ciun 4678  cmpt 4890   I cid 5186  dom cdm 5279  ran crn 5280  cres 5281  ccom 5283  cfv 6070  (class class class)co 6846  0cc0 10193  1c1 10194  0cn0 11542  cuz 11891  t*crtcl 14026  𝑟crelexp 14059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-2 11339  df-n0 11543  df-z 11629  df-uz 11892  df-seq 13014  df-rtrcl 14028  df-relexp 14060
This theorem is referenced by:  brfvrtrcld  38725  fvrtrcllb0d  38726  fvrtrcllb0da  38727  fvrtrcllb1d  38728  dfrtrcl4  38729  corcltrcl  38730  cotrclrcl  38733
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