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Theorem dfrtrcl2 13598
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 2610 . . . 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 5232 . . . . . . . . . . 11 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3 rneq 5258 . . . . . . . . . . 11 (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅)
42, 3uneq12d 3729 . . . . . . . . . 10 (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅))
54reseq2d 5303 . . . . . . . . 9 (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
65sseq1d 3594 . . . . . . . 8 (𝑥 = 𝑅 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧))
7 id 22 . . . . . . . . 9 (𝑥 = 𝑅𝑥 = 𝑅)
87sseq1d 3594 . . . . . . . 8 (𝑥 = 𝑅 → (𝑥𝑧𝑅𝑧))
96, 83anbi12d 1391 . . . . . . 7 (𝑥 = 𝑅 → ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)))
109abbidv 2727 . . . . . 6 (𝑥 = 𝑅 → {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
1110inteqd 4409 . . . . 5 (𝑥 = 𝑅 {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
1211adantl 480 . . . 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 5563 . . . . . . . . . 10 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1614, 15syl 17 . . . . . . . . 9 (𝜑 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1716eqcomd 2615 . . . . . . . 8 (𝜑 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
1814, 13rtrclreclem1 13594 . . . . . . . . 9 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
19 id 22 . . . . . . . . . . 11 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
2019reseq2d 5303 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ 𝑅))
2120sseq1d 3594 . . . . . . . . 9 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ (t*rec‘𝑅)))
2218, 21syl5ibr 234 . . . . . . . 8 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
2317, 22mpcom 37 . . . . . . 7 (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅))
2413rtrclreclem2 13595 . . . . . . 7 (𝜑𝑅 ⊆ (t*rec‘𝑅))
2514, 13rtrclreclem3 13596 . . . . . . 7 (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
26 fvex 6097 . . . . . . . 8 (t*rec‘𝑅) ∈ V
27 sseq2 3589 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
28 sseq2 3589 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → (𝑅𝑧𝑅 ⊆ (t*rec‘𝑅)))
29 id 22 . . . . . . . . . . . . 13 (𝑧 = (t*rec‘𝑅) → 𝑧 = (t*rec‘𝑅))
3029, 29coeq12d 5195 . . . . . . . . . . . 12 (𝑧 = (t*rec‘𝑅) → (𝑧𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅)))
3130, 29sseq12d 3596 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → ((𝑧𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))
3227, 28, 313anbi123d 1390 . . . . . . . . . 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 1841 . . . . . . . 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 3315 . . . . . . . 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 693 . . . . . . 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 1237 . . . . . 6 (𝜑 → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
38 ne0i 3879 . . . . . 6 ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ≠ ∅)
3937, 38syl 17 . . . . 5 (𝜑 → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ≠ ∅)
40 intex 4741 . . . . 5 ({𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ≠ ∅ ↔ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
4139, 40sylib 206 . . . 4 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
421, 12, 13, 41fvmptd 6181 . . 3 (𝜑 → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
43 intss1 4421 . . . . 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))
4437, 43syl 17 . . . 4 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅))
45 vex 3175 . . . . . . . 8 𝑠 ∈ V
46 sseq2 3589 . . . . . . . . 9 (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
47 sseq2 3589 . . . . . . . . 9 (𝑧 = 𝑠 → (𝑅𝑧𝑅𝑠))
48 id 22 . . . . . . . . . . 11 (𝑧 = 𝑠𝑧 = 𝑠)
4948, 48coeq12d 5195 . . . . . . . . . 10 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
5049, 48sseq12d 3596 . . . . . . . . 9 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
5146, 47, 503anbi123d 1390 . . . . . . . 8 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
5245, 51elab 3318 . . . . . . 7 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
5314, 13rtrclreclem4 13597 . . . . . . . 8 (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
545319.21bi 2046 . . . . . . 7 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
5552, 54syl5bi 230 . . . . . 6 (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠))
5655ralrimiv 2947 . . . . 5 (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
57 ssint 4422 . . . . 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)
5856, 57sylibr 222 . . . 4 (𝜑 → (t*rec‘𝑅) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
5944, 58eqssd 3584 . . 3 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (t*rec‘𝑅))
6042, 59eqtrd 2643 . 2 (𝜑 → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))
61 df-rtrcl 13523 . . 3 t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
62 fveq1 6086 . . . . 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))
6362eqeq1d 2611 . . . 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)))
6463imbi2d 328 . . 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))))
6561, 64ax-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)))
6660, 65mpbir 219 1 (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  w3a 1030  wal 1472   = wceq 1474  wcel 1976  {cab 2595  wne 2779  wral 2895  Vcvv 3172  cun 3537  wss 3539  c0 3873   cuni 4366   cint 4404  cmpt 4637   I cid 4937  dom cdm 5027  ran crn 5028  cres 5029  ccom 5031  Rel wrel 5032  cfv 5789  t*crtcl 13521  t*reccrtrcl 13591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-2 10928  df-n0 11142  df-z 11213  df-uz 11522  df-seq 12621  df-rtrcl 13523  df-relexp 13557  df-rtrclrec 13592
This theorem is referenced by:  rtrclind  13601
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