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Theorem rtrclreclem4 15034
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.) (Revised by AV, 13-Jul-2024.)
Hypothesis
Ref Expression
rtrclreclem.1 (𝜑 → Rel 𝑅)
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 2728 . . . . . . 7 ((𝜑𝑅 ∈ V) → (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)))
2 oveq1 7421 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
32iuneq2d 5020 . . . . . . . 8 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
43adantl 481 . . . . . . 7 (((𝜑𝑅 ∈ V) ∧ 𝑟 = 𝑅) → 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
5 simpr 484 . . . . . . 7 ((𝜑𝑅 ∈ V) → 𝑅 ∈ V)
6 nn0ex 12502 . . . . . . . . 9 0 ∈ V
7 ovex 7447 . . . . . . . . 9 (𝑅𝑟𝑛) ∈ V
86, 7iunex 7966 . . . . . . . 8 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
98a1i 11 . . . . . . 7 ((𝜑𝑅 ∈ V) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V)
101, 4, 5, 9fvmptd 7006 . . . . . 6 ((𝜑𝑅 ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
11 eleq1 2816 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 0 → (𝑖 ∈ ℕ0 ↔ 0 ∈ ℕ0))
1211anbi1d 629 . . . . . . . . . . . . . . . . . 18 (𝑖 = 0 → ((𝑖 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ (0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
13 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 0 → (𝑅𝑟𝑖) = (𝑅𝑟0))
1413sseq1d 4009 . . . . . . . . . . . . . . . . . 18 (𝑖 = 0 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟0) ⊆ 𝑠))
1512, 14imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (((𝑖 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑖) ⊆ 𝑠) ↔ ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟0) ⊆ 𝑠)))
16 eleq1 2816 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑚 → (𝑖 ∈ ℕ0𝑚 ∈ ℕ0))
1716anbi1d 629 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑚 → ((𝑖 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ (𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
18 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑚 → (𝑅𝑟𝑖) = (𝑅𝑟𝑚))
1918sseq1d 4009 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑚 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟𝑚) ⊆ 𝑠))
2017, 19imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑚 → (((𝑖 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑖) ⊆ 𝑠) ↔ ((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠)))
21 eleq1 2816 . . . . . . . . . . . . . . . . . . 19 (𝑖 = (𝑚 + 1) → (𝑖 ∈ ℕ0 ↔ (𝑚 + 1) ∈ ℕ0))
2221anbi1d 629 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑚 + 1) → ((𝑖 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
23 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (𝑖 = (𝑚 + 1) → (𝑅𝑟𝑖) = (𝑅𝑟(𝑚 + 1)))
2423sseq1d 4009 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑚 + 1) → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
2522, 24imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑚 + 1) → (((𝑖 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑖) ⊆ 𝑠) ↔ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
26 eleq1 2816 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑛 → (𝑖 ∈ ℕ0𝑛 ∈ ℕ0))
2726anbi1d 629 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑛 → ((𝑖 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ (𝑛 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
28 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑛 → (𝑅𝑟𝑖) = (𝑅𝑟𝑛))
2928sseq1d 4009 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑛 → ((𝑅𝑟𝑖) ⊆ 𝑠 ↔ (𝑅𝑟𝑛) ⊆ 𝑠))
3027, 29imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑛 → (((𝑖 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑖) ⊆ 𝑠) ↔ ((𝑛 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑛) ⊆ 𝑠)))
31 simprll 778 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → 𝜑)
32 rtrclreclem.1 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → Rel 𝑅)
3331, 32syl 17 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → Rel 𝑅)
34 simprlr 779 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → 𝑅 ∈ V)
3533, 34relexp0d 14997 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟0) = ( I ↾ 𝑅))
36 relfld 6273 . . . . . . . . . . . . . . . . . . . 20 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
3733, 36syl 17 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → 𝑅 = (dom 𝑅 ∪ ran 𝑅))
38 simprrr 781 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
3938adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
40 reseq2 5974 . . . . . . . . . . . . . . . . . . . . 21 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ( I ↾ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
4140sseq1d 4009 . . . . . . . . . . . . . . . . . . . 20 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (( I ↾ 𝑅) ⊆ 𝑠 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
4239, 41imbitrrid 245 . . . . . . . . . . . . . . . . . . 19 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ 𝑅) ⊆ 𝑠))
4337, 42mpcom 38 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ 𝑅) ⊆ 𝑠)
4435, 43eqsstrd 4016 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟0) ⊆ 𝑠)
45 simprrr 781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → 𝑚 ∈ ℕ0)
4645adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → 𝑚 ∈ ℕ0)
4746adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → 𝑚 ∈ ℕ0)
4847adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → 𝑚 ∈ ℕ0)
49 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝜑𝑅 ∈ V))
50 simprrl 780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑠𝑠) ⊆ 𝑠)
51 simprrl 780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → 𝑅𝑠)
5251adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → 𝑅𝑠)
53 simprrl 780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
5453adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
5554adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
5650, 52, 55jca32 515 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((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 515 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))))
58 simprrl 780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠))
5958adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → ((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠))
6059adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → ((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠))
6160adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → ((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠))
6257, 61mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑅𝑟𝑚) ⊆ 𝑠)
63 simprll 778 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → 𝜑)
6463adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝜑)
6564, 32syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → Rel 𝑅)
6648adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝑚 ∈ ℕ0)
6765, 66relexpsucrd 15006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
6852adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝑅𝑠)
69 coss2 5853 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑅𝑠 → ((𝑅𝑟𝑚) ∘ 𝑅) ⊆ ((𝑅𝑟𝑚) ∘ 𝑠))
7068, 69syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → ((𝑅𝑟𝑚) ∘ 𝑅) ⊆ ((𝑅𝑟𝑚) ∘ 𝑠))
71 coss1 5852 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑅𝑟𝑚) ⊆ 𝑠 → ((𝑅𝑟𝑚) ∘ 𝑠) ⊆ (𝑠𝑠))
7271, 50sylan9ss 3991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → ((𝑅𝑟𝑚) ∘ 𝑠) ⊆ 𝑠)
7370, 72sstrd 3988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → ((𝑅𝑟𝑚) ∘ 𝑅) ⊆ 𝑠)
7467, 73eqsstrd 4016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑅𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)
7562, 74mpancom 687 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)
7675expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
7776expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → ((𝜑𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
7877expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑅𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝑠𝑠) ⊆ 𝑠 → ((𝜑𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))))
7978anassrs 467 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → ((𝑠𝑠) ⊆ 𝑠 → ((𝜑𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))))
8079impcom 407 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑠𝑠) ⊆ 𝑠 ∧ ((𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝜑𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
8180anassrs 467 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → ((𝜑𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
8281impcom 407 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑅 ∈ V) ∧ (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
8382anassrs 467 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → ((𝑚 + 1) ∈ ℕ0 → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
8483impcom 407 . . . . . . . . . . . . . . . . . . . 20 (((𝑚 + 1) ∈ ℕ0 ∧ (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)
8584anassrs 467 . . . . . . . . . . . . . . . . . . 19 ((((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)
8685expcom 413 . . . . . . . . . . . . . . . . . 18 ((((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠))
8786expcom 413 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → (((𝑚 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑚) ⊆ 𝑠) → (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟(𝑚 + 1)) ⊆ 𝑠)))
8815, 20, 25, 30, 44, 87nn0ind 12681 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑛) ⊆ 𝑠))
8988anabsi5 668 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0 ∧ ((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅𝑟𝑛) ⊆ 𝑠)
9089expcom 413 . . . . . . . . . . . . . 14 (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → (𝑛 ∈ ℕ0 → (𝑅𝑟𝑛) ⊆ 𝑠))
9190ralrimiv 3140 . . . . . . . . . . . . 13 (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ∀𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
92 iunss 5042 . . . . . . . . . . . . 13 ( 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9391, 92sylibr 233 . . . . . . . . . . . 12 (((𝜑𝑅 ∈ V) ∧ ((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9493expcom 413 . . . . . . . . . . 11 (((𝑠𝑠) ⊆ 𝑠 ∧ (𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) → ((𝜑𝑅 ∈ V) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
9594expcom 413 . . . . . . . . . 10 ((𝑅𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) → ((𝑠𝑠) ⊆ 𝑠 → ((𝜑𝑅 ∈ V) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)))
9695expcom 413 . . . . . . . . 9 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 → (𝑅𝑠 → ((𝑠𝑠) ⊆ 𝑠 → ((𝜑𝑅 ∈ V) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))))
97963imp1 1345 . . . . . . . 8 (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ∧ (𝜑𝑅 ∈ V)) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠)
9897expcom 413 . . . . . . 7 ((𝜑𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
99 sseq1 4003 . . . . . . . 8 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ⊆ 𝑠))
10099imbi2d 340 . . . . . . 7 (((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)))
10198, 100imbitrrid 245 . . . . . 6 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → ((𝜑𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠)))
10210, 101mpcom 38 . . . . 5 ((𝜑𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠))
103 df-rtrclrec 15029 . . . . . 6 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
104 fveq1 6890 . . . . . . . . 9 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
105104sseq1d 4009 . . . . . . . 8 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → ((t*rec‘𝑅) ⊆ 𝑠 ↔ ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠))
106105imbi2d 340 . . . . . . 7 (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)))
107106imbi2d 340 . . . . . 6 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (((𝜑𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) ↔ ((𝜑𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠))))
108103, 107ax-mp 5 . . . . 5 (((𝜑𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) ↔ ((𝜑𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) ⊆ 𝑠)))
109102, 108mpbir 230 . . . 4 ((𝜑𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
110109ex 412 . . 3 (𝜑 → (𝑅 ∈ V → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)))
111 fvprc 6883 . . . . 5 𝑅 ∈ V → (t*rec‘𝑅) = ∅)
112 0ss 4392 . . . . 5 ∅ ⊆ 𝑠
113111, 112eqsstrdi 4032 . . . 4 𝑅 ∈ V → (t*rec‘𝑅) ⊆ 𝑠)
114113a1d 25 . . 3 𝑅 ∈ V → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
115110, 114pm2.61d1 180 . 2 (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
116115alrimiv 1923 1 (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1085  wal 1532   = wceq 1534  wcel 2099  wral 3056  Vcvv 3469  cun 3942  wss 3944  c0 4318   cuni 4903   ciun 4991  cmpt 5225   I cid 5569  dom cdm 5672  ran crn 5673  cres 5674  ccom 5676  Rel wrel 5677  cfv 6542  (class class class)co 7414  0cc0 11132  1c1 11133   + caddc 11135  0cn0 12496  𝑟crelexp 14992  t*reccrtrcl 15028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-n0 12497  df-z 12583  df-uz 12847  df-seq 13993  df-relexp 14993  df-rtrclrec 15029
This theorem is referenced by:  dfrtrcl2  15035
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