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Theorem rtrclreclem4 13845
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 2652 . . . . 5 (𝜑 → (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)))
2 oveq1 6697 . . . . . . 7 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
32iuneq2d 4579 . . . . . 6 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
43adantl 481 . . . . 5 ((𝜑𝑟 = 𝑅) → 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
5 rtrclreclem.rex . . . . 5 (𝜑𝑅 ∈ V)
6 nn0ex 11336 . . . . . . 7 0 ∈ V
7 ovex 6718 . . . . . . 7 (𝑅𝑟𝑛) ∈ V
86, 7iunex 7189 . . . . . 6 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
98a1i 11 . . . . 5 (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V)
101, 4, 5, 9fvmptd 6327 . . . 4 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
11 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑖 ∈ ℕ0 ↔ 0 ∈ ℕ0))
1211anbi1d 741 . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑅𝑟𝑖) = (𝑅𝑟0))
1413sseq1d 3665 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟0) ⊆ 𝑠))
1512, 14imbi12d 333 . . . . . . . . . . . . . . 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 2718 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑚 → (𝑖 ∈ ℕ0𝑚 ∈ ℕ0))
1716anbi1d 741 . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑚 → (𝑅𝑟𝑖) = (𝑅𝑟𝑚))
1918sseq1d 3665 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑚 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟𝑚) ⊆ 𝑠))
2017, 19imbi12d 333 . . . . . . . . . . . . . . 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 2718 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚 + 1) → (𝑖 ∈ ℕ0 ↔ (𝑚 + 1) ∈ ℕ0))
2221anbi1d 741 . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚 + 1) → (𝑅𝑟𝑖) = (𝑅𝑟(𝑚 + 1)))
2423sseq1d 3665 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚 + 1) → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
2522, 24imbi12d 333 . . . . . . . . . . . . . . 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 2718 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑛 → (𝑖 ∈ ℕ0𝑛 ∈ ℕ0))
2726anbi1d 741 . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑛 → (𝑅𝑟𝑖) = (𝑅𝑟𝑛))
2928sseq1d 3665 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟𝑛) ⊆ 𝑠))
3027, 29imbi12d 333 . . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → 𝜑)
32 rtrclreclem.rel . . . . . . . . . . . . . . . . . 18 (𝜑 → Rel 𝑅)
3332, 5relexp0d 13808 . . . . . . . . . . . . . . . . 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 5699 . . . . . . . . . . . . . . . . . 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 822 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
3938adantl 481 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
40 reseq2 5423 . . . . . . . . . . . . . . . . . . 19 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ( I ↾ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
4140sseq1d 3665 . . . . . . . . . . . . . . . . . 18 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (( I ↾ 𝑅) ⊆ 𝑠 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
4239, 41syl5ibr 236 . . . . . . . . . . . . . . . . 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 3672 . . . . . . . . . . . . . . 15 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟0) ⊆ 𝑠)
45 simprrr 822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5310 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑅𝑟𝑚) ⊆ 𝑠 → ((𝑅𝑟𝑚) ∘ 𝑠) ⊆ (𝑠𝑠))
7372, 50sylan9ss 3649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3672 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 704 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . . . . 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 681 . . . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . . 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 681 . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . 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 681 . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . 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 681 . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . 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 11510 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑛) ⊆ 𝑠))
9089anabsi5 875 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑛) ⊆ 𝑠)
9190expcom 450 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → (𝑛 ∈ ℕ0 → (𝑅𝑟𝑛) ⊆ 𝑠))
9291ralrimiv 2994 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ∀𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
93 iunss 4593 . . . . . . . . . . 11 ( 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9492, 93sylibr 224 . . . . . . . . . 10 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9594expcom 450 . . . . . . . . 9 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) → (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
9695expcom 450 . . . . . . . 8 ((𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) → ((𝑠𝑠) ⊆ 𝑠 → (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)))
9796expcom 450 . . . . . . 7 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 → (𝑅𝑠 → ((𝑠𝑠) ⊆ 𝑠 → (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))))
98973imp1 1302 . . . . . 6 (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ∧ 𝜑) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9998expcom 450 . . . . 5 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
100 sseq1 3659 . . . . . 6 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
101100imbi2d 329 . . . . 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 236 . . . 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 13840 . . . 4 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
105 fveq1 6228 . . . . . . 7 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
106105sseq1d 3665 . . . . . 6 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → ((t*rec‘𝑅) ⊆ 𝑠 ↔ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠))
107106imbi2d 329 . . . . 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 329 . . . 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 221 . 2 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
111110alrimiv 1895 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 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cun 3605  wss 3607   cuni 4468   ciun 4552  cmpt 4762   I cid 5052  dom cdm 5143  ran crn 5144  cres 5145  ccom 5147  Rel wrel 5148  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  0cn0 11330  𝑟crelexp 13804  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-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-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-relexp 13805  df-rtrclrec 13840
This theorem is referenced by:  dfrtrcl2  13846
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