MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfrtrcl2 Structured version   Visualization version   GIF version

Theorem dfrtrcl2 15098
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.) (Revised by AV, 13-Jul-2024.)
Hypothesis
Ref Expression
dfrtrcl2.1 (𝜑 → Rel 𝑅)
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 2736 . . . . . 6 ((𝜑𝑅 ∈ V) → (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}))
2 dmeq 5917 . . . . . . . . . . . . 13 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3 rneq 5950 . . . . . . . . . . . . 13 (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅)
42, 3uneq12d 4179 . . . . . . . . . . . 12 (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅))
54reseq2d 6000 . . . . . . . . . . 11 (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
65sseq1d 4027 . . . . . . . . . 10 (𝑥 = 𝑅 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧))
7 id 22 . . . . . . . . . . 11 (𝑥 = 𝑅𝑥 = 𝑅)
87sseq1d 4027 . . . . . . . . . 10 (𝑥 = 𝑅 → (𝑥𝑧𝑅𝑧))
96, 83anbi12d 1436 . . . . . . . . 9 (𝑥 = 𝑅 → ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)))
109abbidv 2806 . . . . . . . 8 (𝑥 = 𝑅 → {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
1110inteqd 4956 . . . . . . 7 (𝑥 = 𝑅 {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
1211adantl 481 . . . . . 6 (((𝜑𝑅 ∈ V) ∧ 𝑥 = 𝑅) → {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
13 simpr 484 . . . . . 6 ((𝜑𝑅 ∈ V) → 𝑅 ∈ V)
14 dfrtrcl2.1 . . . . . . . . . . . . 13 (𝜑 → Rel 𝑅)
15 relfld 6297 . . . . . . . . . . . . 13 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1614, 15syl 17 . . . . . . . . . . . 12 (𝜑 𝑅 = (dom 𝑅 ∪ ran 𝑅))
1716eqcomd 2741 . . . . . . . . . . 11 (𝜑 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
1817adantr 480 . . . . . . . . . 10 ((𝜑𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
1914adantr 480 . . . . . . . . . . . 12 ((𝜑𝑅 ∈ V) → Rel 𝑅)
2019, 13rtrclreclem2 15095 . . . . . . . . . . 11 ((𝜑𝑅 ∈ V) → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))
21 id 22 . . . . . . . . . . . . 13 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (dom 𝑅 ∪ ran 𝑅) = 𝑅)
2221reseq2d 6000 . . . . . . . . . . . 12 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ 𝑅))
2322sseq1d 4027 . . . . . . . . . . 11 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ 𝑅) ⊆ (t*rec‘𝑅)))
2420, 23imbitrrid 246 . . . . . . . . . 10 ((dom 𝑅 ∪ ran 𝑅) = 𝑅 → ((𝜑𝑅 ∈ V) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
2518, 24mpcom 38 . . . . . . . . 9 ((𝜑𝑅 ∈ V) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅))
2613rtrclreclem1 15093 . . . . . . . . 9 ((𝜑𝑅 ∈ V) → 𝑅 ⊆ (t*rec‘𝑅))
2714rtrclreclem3 15096 . . . . . . . . . 10 (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
2827adantr 480 . . . . . . . . 9 ((𝜑𝑅 ∈ V) → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
29 fvex 6920 . . . . . . . . . . 11 (t*rec‘𝑅) ∈ V
30 sseq2 4022 . . . . . . . . . . . . . 14 (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)))
31 sseq2 4022 . . . . . . . . . . . . . 14 (𝑧 = (t*rec‘𝑅) → (𝑅𝑧𝑅 ⊆ (t*rec‘𝑅)))
32 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = (t*rec‘𝑅) → 𝑧 = (t*rec‘𝑅))
3332, 32coeq12d 5878 . . . . . . . . . . . . . . 15 (𝑧 = (t*rec‘𝑅) → (𝑧𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅)))
3433, 32sseq12d 4029 . . . . . . . . . . . . . 14 (𝑧 = (t*rec‘𝑅) → ((𝑧𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))
3530, 31, 343anbi123d 1435 . . . . . . . . . . . . 13 (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))))
3635a1i 11 . . . . . . . . . . . 12 (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)))))
3736alrimiv 1925 . . . . . . . . . . 11 (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)))))
38 elabgt 3672 . . . . . . . . . . 11 (((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))))
3929, 37, 38sylancr 587 . . . . . . . . . 10 (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))))
4039adantr 480 . . . . . . . . 9 ((𝜑𝑅 ∈ V) → ((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))))
4125, 26, 28, 40mpbir3and 1341 . . . . . . . 8 ((𝜑𝑅 ∈ V) → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
4241ne0d 4348 . . . . . . 7 ((𝜑𝑅 ∈ V) → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ≠ ∅)
43 intex 5350 . . . . . . 7 ({𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ≠ ∅ ↔ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
4442, 43sylib 218 . . . . . 6 ((𝜑𝑅 ∈ V) → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ∈ V)
451, 12, 13, 44fvmptd 7023 . . . . 5 ((𝜑𝑅 ∈ V) → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
46 intss1 4968 . . . . . . 7 ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅))
4741, 46syl 17 . . . . . 6 ((𝜑𝑅 ∈ V) → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅))
48 vex 3482 . . . . . . . . . . 11 𝑠 ∈ V
49 sseq2 4022 . . . . . . . . . . . 12 (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
50 sseq2 4022 . . . . . . . . . . . 12 (𝑧 = 𝑠 → (𝑅𝑧𝑅𝑠))
51 id 22 . . . . . . . . . . . . . 14 (𝑧 = 𝑠𝑧 = 𝑠)
5251, 51coeq12d 5878 . . . . . . . . . . . . 13 (𝑧 = 𝑠 → (𝑧𝑧) = (𝑠𝑠))
5352, 51sseq12d 4029 . . . . . . . . . . . 12 (𝑧 = 𝑠 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑠𝑠) ⊆ 𝑠))
5449, 50, 533anbi123d 1435 . . . . . . . . . . 11 (𝑧 = 𝑠 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
5548, 54elab 3681 . . . . . . . . . 10 (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠))
5614rtrclreclem4 15097 . . . . . . . . . . 11 (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
575619.21bi 2187 . . . . . . . . . 10 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
5855, 57biimtrid 242 . . . . . . . . 9 (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠))
5958ralrimiv 3143 . . . . . . . 8 (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠)
60 ssint 4969 . . . . . . . 8 ((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)
6159, 60sylibr 234 . . . . . . 7 (𝜑 → (t*rec‘𝑅) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
6261adantr 480 . . . . . 6 ((𝜑𝑅 ∈ V) → (t*rec‘𝑅) ⊆ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
6347, 62eqssd 4013 . . . . 5 ((𝜑𝑅 ∈ V) → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧𝑅𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)} = (t*rec‘𝑅))
6445, 63eqtrd 2775 . . . 4 ((𝜑𝑅 ∈ V) → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))
65 df-rtrcl 15024 . . . . 5 t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
66 fveq1 6906 . . . . . . 7 (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))
6766eqeq1d 2737 . . . . . 6 (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)))
6867imbi2d 340 . . . . 5 (t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}) → (((𝜑𝑅 ∈ V) → (t*‘𝑅) = (t*rec‘𝑅)) ↔ ((𝜑𝑅 ∈ V) → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))))
6965, 68ax-mp 5 . . . 4 (((𝜑𝑅 ∈ V) → (t*‘𝑅) = (t*rec‘𝑅)) ↔ ((𝜑𝑅 ∈ V) → ((𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅)))
7064, 69mpbir 231 . . 3 ((𝜑𝑅 ∈ V) → (t*‘𝑅) = (t*rec‘𝑅))
7170ex 412 . 2 (𝜑 → (𝑅 ∈ V → (t*‘𝑅) = (t*rec‘𝑅)))
72 fvprc 6899 . . 3 𝑅 ∈ V → (t*‘𝑅) = ∅)
73 fvprc 6899 . . . 4 𝑅 ∈ V → (t*rec‘𝑅) = ∅)
7473eqcomd 2741 . . 3 𝑅 ∈ V → ∅ = (t*rec‘𝑅))
7572, 74eqtrd 2775 . 2 𝑅 ∈ V → (t*‘𝑅) = (t*rec‘𝑅))
7671, 75pm2.61d1 180 1 (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  Vcvv 3478  cun 3961  wss 3963  c0 4339   cuni 4912   cint 4951  cmpt 5231   I cid 5582  dom cdm 5689  ran crn 5690  cres 5691  ccom 5693  Rel wrel 5694  cfv 6563  t*crtcl 15022  t*reccrtrcl 15091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040  df-rtrcl 15024  df-relexp 15056  df-rtrclrec 15092
This theorem is referenced by:  rtrclind  15101
  Copyright terms: Public domain W3C validator