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Theorem dfrtrcl2 14025
Description: The two definitions t* and t*rec of the reflexive, transitive closure coincide if 𝑅 is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
Hypotheses
Ref Expression
drrtrcl2.1 (𝜑 → Rel 𝑅)
drrtrcl2.2 (𝜑𝑅 ∈ V)
Assertion
Ref Expression
dfrtrcl2 (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))

Proof of Theorem dfrtrcl2
Dummy variables 𝑥 𝑧 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2807 . . . 4 (𝜑 → (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}))
2 dmeq 5525 . . . . . . . . . . 11 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3 rneq 5552 . . . . . . . . . . 11 (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅)
42, 3uneq12d 3967 . . . . . . . . . 10 (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅))
54reseq2d 5597 . . . . . . . . 9 (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
65sseq1d 3829 . . . . . . . 8 (𝑥 = 𝑅 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧))
7 id 22 . . . . . . . . 9 (𝑥 = 𝑅𝑥 = 𝑅)
87sseq1d 3829 . . . . . . . 8 (𝑥 = 𝑅 → (𝑥𝑧𝑅𝑧))
96, 83anbi12d 1554 . . . . . . 7 (𝑥 = 𝑅 → ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)))
109abbidv 2925 . . . . . 6 (𝑥 = 𝑅 → {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
1110inteqd 4674 . . . . 5 (𝑥 = 𝑅 {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
1211adantl 469 . . . 4 ((𝜑𝑥 = 𝑅) → {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
13 drrtrcl2.2 . . . 4 (𝜑𝑅 ∈ V)
14 drrtrcl2.1 . . . . . . . . . 10 (𝜑 → Rel 𝑅)
15 relfld 5875 . . . . . . . . . 10 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1614, 15syl 17 . . . . . . . . 9 (𝜑 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1716eqcomd 2812 . . . . . . . 8 (𝜑 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
1814, 13rtrclreclem1 14021 . . . . . . . . 9 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
19 id 22 . . . . . . . . . . 11 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
2019reseq2d 5597 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ 𝑅))
2120sseq1d 3829 . . . . . . . . 9 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ (t*rec‘𝑅)))
2218, 21syl5ibr 237 . . . . . . . 8 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
2317, 22mpcom 38 . . . . . . 7 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅))
2413rtrclreclem2 14022 . . . . . . 7 (𝜑𝑅 ⊆ (t*rec‘𝑅))
2514, 13rtrclreclem3 14023 . . . . . . 7 (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
26 fvex 6421 . . . . . . . 8 (t*rec‘𝑅) ∈ V
27 sseq2 3824 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
28 sseq2 3824 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → (𝑅𝑧𝑅 ⊆ (t*rec‘𝑅)))
29 id 22 . . . . . . . . . . . . 13 (𝑧 = (t*rec‘𝑅) → 𝑧 = (t*rec‘𝑅))
3029, 29coeq12d 5488 . . . . . . . . . . . 12 (𝑧 = (t*rec‘𝑅) → (𝑧𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅)))
3130, 29sseq12d 3831 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → ((𝑧𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))
3227, 28, 313anbi123d 1553 . . . . . . . . . 10 (𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))
3332a1i 11 . . . . . . . . 9 (𝜑 → (𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))))
3433alrimiv 2018 . . . . . . . 8 (𝜑 → ∀𝑧(𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))))
35 elabgt 3542 . . . . . . . 8 (((t*rec‘𝑅) ∈ V ∧ ∀𝑧(𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))) → ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))
3626, 34, 35sylancr 577 . . . . . . 7 (𝜑 → ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))
3723, 24, 25, 36mpbir3and 1435 . . . . . 6 (𝜑 → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
3837ne0d 4123 . . . . 5 (𝜑 → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ≠ ∅)
39 intex 5012 . . . . 5 ({𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ≠ ∅ ↔ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
4038, 39sylib 209 . . . 4 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
411, 12, 13, 40fvmptd 6509 . . 3 (𝜑 → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
42 intss1 4684 . . . . 5 ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅))
4337, 42syl 17 . . . 4 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅))
44 vex 3394 . . . . . . . 8 𝑠 ∈ V
45 sseq2 3824 . . . . . . . . 9 (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
46 sseq2 3824 . . . . . . . . 9 (𝑧 = 𝑠 → (𝑅𝑧𝑅𝑠))
47 id 22 . . . . . . . . . . 11 (𝑧 = 𝑠𝑧 = 𝑠)
4847, 47coeq12d 5488 . . . . . . . . . 10 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
4948, 47sseq12d 3831 . . . . . . . . 9 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
5045, 46, 493anbi123d 1553 . . . . . . . 8 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
5144, 50elab 3545 . . . . . . 7 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
5214, 13rtrclreclem4 14024 . . . . . . . 8 (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
535219.21bi 2224 . . . . . . 7 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
5451, 53syl5bi 233 . . . . . 6 (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠))
5554ralrimiv 3153 . . . . 5 (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
56 ssint 4685 . . . . 5 ((t*rec‘𝑅) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
5755, 56sylibr 225 . . . 4 (𝜑 → (t*rec‘𝑅) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
5843, 57eqssd 3815 . . 3 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (t*rec‘𝑅))
5941, 58eqtrd 2840 . 2 (𝜑 → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))
60 df-rtrcl 13952 . . 3 t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
61 fveq1 6407 . . . . 5 (t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) → (t*‘𝑅) = ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅))
6261eqeq1d 2808 . . . 4 (t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) → ((t*‘𝑅) = (t*rec‘𝑅) ↔ ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅)))
6362imbi2d 331 . . 3 (t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) → ((𝜑 → (t*‘𝑅) = (t*rec‘𝑅)) ↔ (𝜑 → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))))
6460, 63ax-mp 5 . 2 ((𝜑 → (t*‘𝑅) = (t*rec‘𝑅)) ↔ (𝜑 → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅)))
6559, 64mpbir 222 1 (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  w3a 1100  wal 1635   = wceq 1637  wcel 2156  {cab 2792  wne 2978  wral 3096  Vcvv 3391  cun 3767  wss 3769  c0 4116   cuni 4630   cint 4669  cmpt 4923   I cid 5218  dom cdm 5311  ran crn 5312  cres 5313  ccom 5315  Rel wrel 5316  cfv 6101  t*crtcl 13950  t*reccrtrcl 14018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-er 7979  df-en 8193  df-dom 8194  df-sdom 8195  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-nn 11306  df-2 11364  df-n0 11560  df-z 11644  df-uz 11905  df-seq 13025  df-rtrcl 13952  df-relexp 13984  df-rtrclrec 14019
This theorem is referenced by:  rtrclind  14028
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