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Theorem rtrclreclem4 14100
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 2766 . . . . 5 (𝜑 → (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)))
2 oveq1 6853 . . . . . . 7 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
32iuneq2d 4705 . . . . . 6 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
43adantl 473 . . . . 5 ((𝜑𝑟 = 𝑅) → 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
5 rtrclreclem.rex . . . . 5 (𝜑𝑅 ∈ V)
6 nn0ex 11549 . . . . . . 7 0 ∈ V
7 ovex 6878 . . . . . . 7 (𝑅𝑟𝑛) ∈ V
86, 7iunex 7349 . . . . . 6 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
98a1i 11 . . . . 5 (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V)
101, 4, 5, 9fvmptd 6481 . . . 4 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
11 eleq1 2832 . . . . . . . . . . . . . . . . 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 6854 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑅𝑟𝑖) = (𝑅𝑟0))
1413sseq1d 3794 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟0) ⊆ 𝑠))
1512, 14imbi12d 335 . . . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . . . 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 6854 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑚 → (𝑅𝑟𝑖) = (𝑅𝑟𝑚))
1918sseq1d 3794 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑚 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟𝑚) ⊆ 𝑠))
2017, 19imbi12d 335 . . . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . . . 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 6854 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚 + 1) → (𝑅𝑟𝑖) = (𝑅𝑟(𝑚 + 1)))
2423sseq1d 3794 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑚 + 1) → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
2522, 24imbi12d 335 . . . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . . . 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 6854 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑛 → (𝑅𝑟𝑖) = (𝑅𝑟𝑛))
2928sseq1d 3794 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟𝑛) ⊆ 𝑠))
3027, 29imbi12d 335 . . . . . . . . . . . . . . 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 14063 . . . . . . . . . . . . . . . . 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 5849 . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ (𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
40 reseq2 5562 . . . . . . . . . . . . . . . . . . 19 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ( I ↾ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
4140sseq1d 3794 . . . . . . . . . . . . . . . . . 18 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (( I ↾ 𝑅) ⊆ 𝑠 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
4239, 41syl5ibr 237 . . . . . . . . . . . . . . . . 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 3801 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14069 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑅𝑟𝑚) ⊆ 𝑠 → ((𝑅𝑟𝑚) ∘ 𝑠) ⊆ (𝑠𝑠))
7372, 50sylan9ss 3776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3801 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 402 . . . . . . . . . . . . . . . . . . . . . . . . . 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 402 . . . . . . . . . . . . . . . . . . . . . . . . 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 402 . . . . . . . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . . . . . 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 396 . . . . . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . . . 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 396 . . . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . 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 396 . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . 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 402 . . . . . . . . . . . . . . . 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 402 . . . . . . . . . . . . . . 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 11724 . . . . . . . . . . . . . 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 402 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → (𝑛 ∈ ℕ0 → (𝑅𝑟𝑛) ⊆ 𝑠))
9291ralrimiv 3112 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ∀𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
93 iunss 4719 . . . . . . . . . . 11 ( 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9492, 93sylibr 225 . . . . . . . . . 10 ((𝜑 ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9594expcom 402 . . . . . . . . 9 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) → (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
9695expcom 402 . . . . . . . 8 ((𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) → ((𝑠𝑠) ⊆ 𝑠 → (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)))
9796expcom 402 . . . . . . 7 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 → (𝑅𝑠 → ((𝑠𝑠) ⊆ 𝑠 → (𝜑 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))))
98973imp1 1456 . . . . . 6 (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ∧ 𝜑) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9998expcom 402 . . . . 5 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
100 sseq1 3788 . . . . . 6 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
101100imbi2d 331 . . . . 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 237 . . . 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 14095 . . . 4 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
105 fveq1 6378 . . . . . . 7 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
106105sseq1d 3794 . . . . . 6 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → ((t*rec‘𝑅) ⊆ 𝑠 ↔ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠))
107106imbi2d 331 . . . . 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 331 . . . 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 222 . 2 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
111110alrimiv 2022 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 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cun 3732  wss 3734   cuni 4596   ciun 4678  cmpt 4890   I cid 5186  dom cdm 5279  ran crn 5280  cres 5281  ccom 5283  Rel wrel 5284  cfv 6070  (class class class)co 6846  0cc0 10193  1c1 10194   + caddc 10196  0cn0 11542  𝑟crelexp 14059  t*reccrtrcl 14094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-n0 11543  df-z 11629  df-uz 11892  df-seq 13014  df-relexp 14060  df-rtrclrec 14095
This theorem is referenced by:  dfrtrcl2  14101
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