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Theorem dfrtrcl2 13846
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 2652 . . . 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 5356 . . . . . . . . . . 11 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3 rneq 5383 . . . . . . . . . . 11 (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅)
42, 3uneq12d 3801 . . . . . . . . . 10 (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅))
54reseq2d 5428 . . . . . . . . 9 (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
65sseq1d 3665 . . . . . . . 8 (𝑥 = 𝑅 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧))
7 id 22 . . . . . . . . 9 (𝑥 = 𝑅𝑥 = 𝑅)
87sseq1d 3665 . . . . . . . 8 (𝑥 = 𝑅 → (𝑥𝑧𝑅𝑧))
96, 83anbi12d 1440 . . . . . . 7 (𝑥 = 𝑅 → ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)))
109abbidv 2770 . . . . . 6 (𝑥 = 𝑅 → {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
1110inteqd 4512 . . . . 5 (𝑥 = 𝑅 {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
1211adantl 481 . . . 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 5699 . . . . . . . . . 10 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1614, 15syl 17 . . . . . . . . 9 (𝜑 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1716eqcomd 2657 . . . . . . . 8 (𝜑 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
1814, 13rtrclreclem1 13842 . . . . . . . . 9 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
19 id 22 . . . . . . . . . . 11 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
2019reseq2d 5428 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ 𝑅))
2120sseq1d 3665 . . . . . . . . 9 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ (t*rec‘𝑅)))
2218, 21syl5ibr 236 . . . . . . . 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 13843 . . . . . . 7 (𝜑𝑅 ⊆ (t*rec‘𝑅))
2514, 13rtrclreclem3 13844 . . . . . . 7 (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
26 fvex 6239 . . . . . . . 8 (t*rec‘𝑅) ∈ V
27 sseq2 3660 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
28 sseq2 3660 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → (𝑅𝑧𝑅 ⊆ (t*rec‘𝑅)))
29 id 22 . . . . . . . . . . . . 13 (𝑧 = (t*rec‘𝑅) → 𝑧 = (t*rec‘𝑅))
3029, 29coeq12d 5319 . . . . . . . . . . . 12 (𝑧 = (t*rec‘𝑅) → (𝑧𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅)))
3130, 29sseq12d 3667 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → ((𝑧𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))
3227, 28, 313anbi123d 1439 . . . . . . . . . 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 1895 . . . . . . . 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 3379 . . . . . . . 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 696 . . . . . . 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 1264 . . . . . 6 (𝜑 → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
38 ne0i 3954 . . . . . 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 4850 . . . . 5 ({𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ≠ ∅ ↔ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
4139, 40sylib 208 . . . 4 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
421, 12, 13, 41fvmptd 6327 . . 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 4524 . . . . 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 3234 . . . . . . . 8 𝑠 ∈ V
46 sseq2 3660 . . . . . . . . 9 (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
47 sseq2 3660 . . . . . . . . 9 (𝑧 = 𝑠 → (𝑅𝑧𝑅𝑠))
48 id 22 . . . . . . . . . . 11 (𝑧 = 𝑠𝑧 = 𝑠)
4948, 48coeq12d 5319 . . . . . . . . . 10 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
5049, 48sseq12d 3667 . . . . . . . . 9 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
5146, 47, 503anbi123d 1439 . . . . . . . 8 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
5245, 51elab 3382 . . . . . . 7 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
5314, 13rtrclreclem4 13845 . . . . . . . 8 (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
545319.21bi 2097 . . . . . . 7 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
5552, 54syl5bi 232 . . . . . 6 (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠))
5655ralrimiv 2994 . . . . 5 (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
57 ssint 4525 . . . . 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 224 . . . 4 (𝜑 → (t*rec‘𝑅) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
5944, 58eqssd 3653 . . 3 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (t*rec‘𝑅))
6042, 59eqtrd 2685 . 2 (𝜑 → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))
61 df-rtrcl 13773 . . 3 t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
62 fveq1 6228 . . . . 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 2653 . . . 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 329 . . 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 221 1 (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  Vcvv 3231  cun 3605  wss 3607  c0 3948   cuni 4468   cint 4507  cmpt 4762   I cid 5052  dom cdm 5143  ran crn 5144  cres 5145  ccom 5147  Rel wrel 5148  cfv 5926  t*crtcl 13771  t*reccrtrcl 13839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-rtrcl 13773  df-relexp 13805  df-rtrclrec 13840
This theorem is referenced by:  rtrclind  13849
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