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Theorem dfrtrcl2 13650
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 2611 . . . 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 5246 . . . . . . . . . . 11 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3 rneq 5272 . . . . . . . . . . 11 (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅)
42, 3uneq12d 3730 . . . . . . . . . 10 (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅))
54reseq2d 5317 . . . . . . . . 9 (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
65sseq1d 3595 . . . . . . . 8 (𝑥 = 𝑅 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧))
7 id 22 . . . . . . . . 9 (𝑥 = 𝑅𝑥 = 𝑅)
87sseq1d 3595 . . . . . . . 8 (𝑥 = 𝑅 → (𝑥𝑧𝑅𝑧))
96, 83anbi12d 1392 . . . . . . 7 (𝑥 = 𝑅 → ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)))
109abbidv 2728 . . . . . 6 (𝑥 = 𝑅 → {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
1110inteqd 4415 . . . . 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 5578 . . . . . . . . . 10 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1614, 15syl 17 . . . . . . . . 9 (𝜑 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1716eqcomd 2616 . . . . . . . 8 (𝜑 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
1814, 13rtrclreclem1 13646 . . . . . . . . 9 (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
19 id 22 . . . . . . . . . . 11 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
2019reseq2d 5317 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ 𝑅))
2120sseq1d 3595 . . . . . . . . 9 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ (t*rec‘𝑅)))
2218, 21syl5ibr 235 . . . . . . . 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 13647 . . . . . . 7 (𝜑𝑅 ⊆ (t*rec‘𝑅))
2514, 13rtrclreclem3 13648 . . . . . . 7 (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
26 fvex 6113 . . . . . . . 8 (t*rec‘𝑅) ∈ V
27 sseq2 3590 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
28 sseq2 3590 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → (𝑅𝑧𝑅 ⊆ (t*rec‘𝑅)))
29 id 22 . . . . . . . . . . . . 13 (𝑧 = (t*rec‘𝑅) → 𝑧 = (t*rec‘𝑅))
3029, 29coeq12d 5208 . . . . . . . . . . . 12 (𝑧 = (t*rec‘𝑅) → (𝑧𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅)))
3130, 29sseq12d 3597 . . . . . . . . . . 11 (𝑧 = (t*rec‘𝑅) → ((𝑧𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))
3227, 28, 313anbi123d 1391 . . . . . . . . . 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 1842 . . . . . . . 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 3316 . . . . . . . 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 694 . . . . . . 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 1238 . . . . . 6 (𝜑 → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
38 ne0i 3880 . . . . . 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 4747 . . . . 5 ({𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ≠ ∅ ↔ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
4139, 40sylib 207 . . . 4 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
421, 12, 13, 41fvmptd 6197 . . 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 4427 . . . . 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 3176 . . . . . . . 8 𝑠 ∈ V
46 sseq2 3590 . . . . . . . . 9 (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
47 sseq2 3590 . . . . . . . . 9 (𝑧 = 𝑠 → (𝑅𝑧𝑅𝑠))
48 id 22 . . . . . . . . . . 11 (𝑧 = 𝑠𝑧 = 𝑠)
4948, 48coeq12d 5208 . . . . . . . . . 10 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
5049, 48sseq12d 3597 . . . . . . . . 9 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
5146, 47, 503anbi123d 1391 . . . . . . . 8 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
5245, 51elab 3319 . . . . . . 7 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
5314, 13rtrclreclem4 13649 . . . . . . . 8 (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
545319.21bi 2047 . . . . . . 7 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
5552, 54syl5bi 231 . . . . . 6 (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠))
5655ralrimiv 2948 . . . . 5 (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
57 ssint 4428 . . . . 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 223 . . . 4 (𝜑 → (t*rec‘𝑅) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
5944, 58eqssd 3585 . . 3 (𝜑 {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (t*rec‘𝑅))
6042, 59eqtrd 2644 . 2 (𝜑 → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))
61 df-rtrcl 13575 . . 3 t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
62 fveq1 6102 . . . . 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 2612 . . . 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 220 1 (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  w3a 1031  wal 1473   = wceq 1475  wcel 1977  {cab 2596  wne 2780  wral 2896  Vcvv 3173  cun 3538  wss 3540  c0 3874   cuni 4372   cint 4410  cmpt 4643   I cid 4948  dom cdm 5038  ran crn 5039  cres 5040  ccom 5042  Rel wrel 5043  cfv 5804  t*crtcl 13573  t*reccrtrcl 13643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-rtrcl 13575  df-relexp 13609  df-rtrclrec 13644
This theorem is referenced by:  rtrclind  13653
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