MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rtrclreclem4 Structured version   Visualization version   GIF version

Theorem rtrclreclem4 14178
Description: The reflexive, transitive closure of 𝑅 is the smallest reflexive, transitive relation which contains 𝑅 and the identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
Hypotheses
Ref Expression
rtrclreclem.rel (𝜑 → Rel 𝑅)
rtrclreclem.rex (𝜑𝑅 ∈ V)
Assertion
Ref Expression
rtrclreclem4 (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
Distinct variable group:   𝜑,𝑠
Allowed substitution hint:   𝑅(𝑠)

Proof of Theorem rtrclreclem4
Dummy variables 𝑛 𝑖 𝑚 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2826 . . . . 5 (𝜑 → (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)))
2 oveq1 6912 . . . . . . 7 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
32iuneq2d 4767 . . . . . 6 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
43adantl 475 . . . . 5 ((𝜑𝑟 = 𝑅) → 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
5 rtrclreclem.rex . . . . 5 (𝜑𝑅 ∈ V)
6 nn0ex 11625 . . . . . . 7 0 ∈ V
7 ovex 6937 . . . . . . 7 (𝑅𝑟𝑛) ∈ V
86, 7iunex 7408 . . . . . 6 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
98a1i 11 . . . . 5 (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V)
101, 4, 5, 9fvmptd 6535 . . . 4 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
11 eleq1 2894 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑖 ∈ ℕ0 ↔ 0 ∈ ℕ0))
1211anbi1d 623 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → ((𝑖 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ (0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
13 oveq2 6913 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑅𝑟𝑖) = (𝑅𝑟0))
1413sseq1d 3857 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟0) ⊆ 𝑠))
1512, 14imbi12d 336 . . . . . . . . . . . . . . 15 (𝑖 = 0 → (((𝑖 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑖) ⊆ 𝑠) ↔ ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟0) ⊆ 𝑠)))
16 eleq1 2894 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑚 → (𝑖 ∈ ℕ0𝑚 ∈ ℕ0))
1716anbi1d 623 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑚 → ((𝑖 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
18 oveq2 6913 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑚 → (𝑅𝑟𝑖) = (𝑅𝑟𝑚))
1918sseq1d 3857 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑚 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟𝑚) ⊆ 𝑠))
2017, 19imbi12d 336 . . . . . . . . . . . . . . 15 (𝑖 = 𝑚 → (((𝑖 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑖) ⊆ 𝑠) ↔ ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠)))
21 eleq1 2894 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚 + 1) → (𝑖 ∈ ℕ0 ↔ (𝑚 + 1) ∈ ℕ0))
2221anbi1d 623 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚 + 1) → ((𝑖 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
23 oveq2 6913 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚 + 1) → (𝑅𝑟𝑖) = (𝑅𝑟(𝑚 + 1)))
2423sseq1d 3857 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚 + 1) → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
2522, 24imbi12d 336 . . . . . . . . . . . . . . 15 (𝑖 = (𝑚 + 1) → (((𝑖 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑖) ⊆ 𝑠) ↔ (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
26 eleq1 2894 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑛 → (𝑖 ∈ ℕ0𝑛 ∈ ℕ0))
2726anbi1d 623 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → ((𝑖 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ (𝑛 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
28 oveq2 6913 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑛 → (𝑅𝑟𝑖) = (𝑅𝑟𝑛))
2928sseq1d 3857 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟𝑛) ⊆ 𝑠))
3027, 29imbi12d 336 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → (((𝑖 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑖) ⊆ 𝑠) ↔ ((𝑛 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑛) ⊆ 𝑠)))
31 simprl 787 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → 𝜑)
32 rtrclreclem.rel . . . . . . . . . . . . . . . . . 18 (𝜑 → Rel 𝑅)
3332, 5relexp0d 14141 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
3431, 33syl 17 . . . . . . . . . . . . . . . 16 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟0) = ( I ↾ 𝑅))
3531, 32syl 17 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → Rel 𝑅)
36 relfld 5902 . . . . . . . . . . . . . . . . . 18 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
3735, 36syl 17 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → 𝑅 = (dom 𝑅 ∪ ran 𝑅))
38 simprrr 800 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
3938adantl 475 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
40 reseq2 5624 . . . . . . . . . . . . . . . . . . 19 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ( I ↾ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
4140sseq1d 3857 . . . . . . . . . . . . . . . . . 18 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (( I ↾ 𝑅) ⊆ 𝑠 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
4239, 41syl5ibr 238 . . . . . . . . . . . . . . . . 17 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ 𝑅) ⊆ 𝑠))
4337, 42mpcom 38 . . . . . . . . . . . . . . . 16 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ 𝑅) ⊆ 𝑠)
4434, 43eqsstrd 3864 . . . . . . . . . . . . . . 15 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟0) ⊆ 𝑠)
45 simprrr 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → 𝑚 ∈ ℕ0)
4645adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → 𝑚 ∈ ℕ0)
4746adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → 𝑚 ∈ ℕ0)
4847adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → 𝑚 ∈ ℕ0)
49 simprl 787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → 𝜑)
50 simprrl 799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑠𝑠) ⊆ 𝑠)
51 simprrl 799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → 𝑅𝑠)
5251adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → 𝑅𝑠)
53 simprrl 799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
5453adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
5554adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
5650, 52, 55jca32 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))
5748, 49, 56jca32 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))))
58 simprrl 799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠))
5958adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠))
6059adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠))
6160adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → ((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠))
6257, 61mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑅𝑟𝑚) ⊆ 𝑠)
6348adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝑚 ∈ ℕ0)
64 simprrl 799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝜑)
6564, 32syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → Rel 𝑅)
6664, 5syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝑅 ∈ V)
6765, 66relexpsucrd 14147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → (𝑚 ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅)))
6863, 67mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
6952adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝑅𝑠)
70 coss2 5511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑅𝑠 → ((𝑅𝑟𝑚) ∘ 𝑅) ⊆ ((𝑅𝑟𝑚) ∘ 𝑠))
7169, 70syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → ((𝑅𝑟𝑚) ∘ 𝑅) ⊆ ((𝑅𝑟𝑚) ∘ 𝑠))
72 coss1 5510 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑅𝑟𝑚) ⊆ 𝑠 → ((𝑅𝑟𝑚) ∘ 𝑠) ⊆ (𝑠𝑠))
7372, 50sylan9ss 3840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → ((𝑅𝑟𝑚) ∘ 𝑠) ⊆ 𝑠)
7471, 73sstrd 3837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → ((𝑅𝑟𝑚) ∘ 𝑅) ⊆ 𝑠)
7568, 74eqsstrd 3864 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)
7662, 75mpancom 679 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)
7776expcom 404 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
7877expcom 404 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → (𝜑 → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
7978expcom 404 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝑠𝑠) ⊆ 𝑠 → (𝜑 → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))))
8079anassrs 461 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → ((𝑠𝑠) ⊆ 𝑠 → (𝜑 → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))))
8180impcom 398 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑠𝑠) ⊆ 𝑠 ∧ ((𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → (𝜑 → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
8281anassrs 461 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → (𝜑 → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
8382impcom 398 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
8483anassrs 461 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
8584impcom 398 . . . . . . . . . . . . . . . . . 18 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)
8685anassrs 461 . . . . . . . . . . . . . . . . 17 ((((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ∧ (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)
8786expcom 404 . . . . . . . . . . . . . . . 16 ((((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
8887expcom 404 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → (((𝑚 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) → (((𝑚 + 1) ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
8915, 20, 25, 30, 44, 88nn0ind 11800 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑛) ⊆ 𝑠))
9089anabsi5 659 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑛) ⊆ 𝑠)
9190expcom 404 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → (𝑛 ∈ ℕ0 → (𝑅𝑟𝑛) ⊆ 𝑠))
9291ralrimiv 3174 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ∀𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
93 iunss 4781 . . . . . . . . . . 11 ( 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9492, 93sylibr 226 . . . . . . . . . 10 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9594expcom 404 . . . . . . . . 9 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) → (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
9695expcom 404 . . . . . . . 8 ((𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) → ((𝑠𝑠) ⊆ 𝑠 → (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)))
9796expcom 404 . . . . . . 7 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 → (𝑅𝑠 → ((𝑠𝑠) ⊆ 𝑠 → (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))))
98973imp1 1460 . . . . . 6 (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ∧ 𝜑) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9998expcom 404 . . . . 5 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
100 sseq1 3851 . . . . . 6 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
101100imbi2d 332 . . . . 5 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠) ↔ ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)))
10299, 101syl5ibr 238 . . . 4 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠)))
10310, 102mpcom 38 . . 3 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠))
104 df-rtrclrec 14173 . . . 4 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
105 fveq1 6432 . . . . . . 7 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
106105sseq1d 3857 . . . . . 6 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → ((t*rec‘𝑅) ⊆ 𝑠 ↔ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠))
107106imbi2d 332 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠) ↔ ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠)))
108107imbi2d 332 . . . 4 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → ((𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) ↔ (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠))))
109104, 108ax-mp 5 . . 3 ((𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) ↔ (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠)))
110103, 109mpbir 223 . 2 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
111110alrimiv 2026 1 (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111  wal 1654   = wceq 1656  wcel 2164  wral 3117  Vcvv 3414  cun 3796  wss 3798   cuni 4658   ciun 4740  cmpt 4952   I cid 5249  dom cdm 5342  ran crn 5343  cres 5344  ccom 5346  Rel wrel 5347  cfv 6123  (class class class)co 6905  0cc0 10252  1c1 10253   + caddc 10255  0cn0 11618  𝑟crelexp 14137  t*reccrtrcl 14172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-n0 11619  df-z 11705  df-uz 11969  df-seq 13096  df-relexp 14138  df-rtrclrec 14173
This theorem is referenced by:  dfrtrcl2  14179
  Copyright terms: Public domain W3C validator