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Theorem noseqrdgsuc 28329
Description: Successor value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 19-Apr-2025.)
Hypotheses
Ref Expression
om2noseq.1 (𝜑𝐶 No )
om2noseq.2 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseq.3 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
noseqrdg.1 (𝜑𝐴𝑉)
noseqrdg.2 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
noseqrdg.3 (𝜑𝑆 = ran 𝑅)
Assertion
Ref Expression
noseqrdgsuc ((𝜑𝐵𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆𝐵)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐹,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem noseqrdgsuc
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 om2noseq.1 . . . . . . 7 (𝜑𝐶 No )
2 om2noseq.2 . . . . . . 7 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
3 om2noseq.3 . . . . . . 7 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
4 noseqrdg.1 . . . . . . 7 (𝜑𝐴𝑉)
5 noseqrdg.2 . . . . . . 7 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
6 noseqrdg.3 . . . . . . 7 (𝜑𝑆 = ran 𝑅)
71, 2, 3, 4, 5, 6noseqrdgfn 28327 . . . . . 6 (𝜑𝑆 Fn 𝑍)
87adantr 480 . . . . 5 ((𝜑𝐵𝑍) → 𝑆 Fn 𝑍)
98fnfund 6670 . . . 4 ((𝜑𝐵𝑍) → Fun 𝑆)
103adantr 480 . . . . . . 7 ((𝜑𝐵𝑍) → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
111adantr 480 . . . . . . 7 ((𝜑𝐵𝑍) → 𝐶 No )
12 simpr 484 . . . . . . 7 ((𝜑𝐵𝑍) → 𝐵𝑍)
1310, 11, 12noseqp1 28312 . . . . . 6 ((𝜑𝐵𝑍) → (𝐵 +s 1s ) ∈ 𝑍)
141, 2, 3, 4, 5noseqrdglem 28326 . . . . . 6 ((𝜑 ∧ (𝐵 +s 1s ) ∈ 𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))⟩ ∈ ran 𝑅)
1513, 14syldan 591 . . . . 5 ((𝜑𝐵𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))⟩ ∈ ran 𝑅)
166adantr 480 . . . . 5 ((𝜑𝐵𝑍) → 𝑆 = ran 𝑅)
1715, 16eleqtrrd 2842 . . . 4 ((𝜑𝐵𝑍) → ⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))⟩ ∈ 𝑆)
18 funopfv 6959 . . . 4 (Fun 𝑆 → (⟨(𝐵 +s 1s ), (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))⟩ ∈ 𝑆 → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s ))))))
199, 17, 18sylc 65 . . 3 ((𝜑𝐵𝑍) → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s )))))
201, 2, 3om2noseqf1o 28322 . . . . . . . 8 (𝜑𝐺:ω–1-1-onto𝑍)
2120adantr 480 . . . . . . 7 ((𝜑𝐵𝑍) → 𝐺:ω–1-1-onto𝑍)
22 f1ocnvdm 7305 . . . . . . . . 9 ((𝐺:ω–1-1-onto𝑍𝐵𝑍) → (𝐺𝐵) ∈ ω)
2320, 22sylan 580 . . . . . . . 8 ((𝜑𝐵𝑍) → (𝐺𝐵) ∈ ω)
24 peano2 7913 . . . . . . . 8 ((𝐺𝐵) ∈ ω → suc (𝐺𝐵) ∈ ω)
2523, 24syl 17 . . . . . . 7 ((𝜑𝐵𝑍) → suc (𝐺𝐵) ∈ ω)
2621, 25jca 511 . . . . . 6 ((𝜑𝐵𝑍) → (𝐺:ω–1-1-onto𝑍 ∧ suc (𝐺𝐵) ∈ ω))
272adantr 480 . . . . . . . 8 ((𝜑𝐵𝑍) → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
2811, 27, 23om2noseqsuc 28318 . . . . . . 7 ((𝜑𝐵𝑍) → (𝐺‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵)) +s 1s ))
29 f1ocnvfv2 7297 . . . . . . . . 9 ((𝐺:ω–1-1-onto𝑍𝐵𝑍) → (𝐺‘(𝐺𝐵)) = 𝐵)
3020, 29sylan 580 . . . . . . . 8 ((𝜑𝐵𝑍) → (𝐺‘(𝐺𝐵)) = 𝐵)
3130oveq1d 7446 . . . . . . 7 ((𝜑𝐵𝑍) → ((𝐺‘(𝐺𝐵)) +s 1s ) = (𝐵 +s 1s ))
3228, 31eqtrd 2775 . . . . . 6 ((𝜑𝐵𝑍) → (𝐺‘suc (𝐺𝐵)) = (𝐵 +s 1s ))
33 f1ocnvfv 7298 . . . . . 6 ((𝐺:ω–1-1-onto𝑍 ∧ suc (𝐺𝐵) ∈ ω) → ((𝐺‘suc (𝐺𝐵)) = (𝐵 +s 1s ) → (𝐺‘(𝐵 +s 1s )) = suc (𝐺𝐵)))
3426, 32, 33sylc 65 . . . . 5 ((𝜑𝐵𝑍) → (𝐺‘(𝐵 +s 1s )) = suc (𝐺𝐵))
3534fveq2d 6911 . . . 4 ((𝜑𝐵𝑍) → (𝑅‘(𝐺‘(𝐵 +s 1s ))) = (𝑅‘suc (𝐺𝐵)))
3635fveq2d 6911 . . 3 ((𝜑𝐵𝑍) → (2nd ‘(𝑅‘(𝐺‘(𝐵 +s 1s )))) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
3719, 36eqtrd 2775 . 2 ((𝜑𝐵𝑍) → (𝑆‘(𝐵 +s 1s )) = (2nd ‘(𝑅‘suc (𝐺𝐵))))
38 frsuc 8476 . . . . . . . . 9 ((𝐺𝐵) ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))))
3938adantl 481 . . . . . . . 8 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))))
405fveq1d 6909 . . . . . . . . 9 (𝜑 → (𝑅‘suc (𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵)))
4140adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘suc (𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘suc (𝐺𝐵)))
425fveq1d 6909 . . . . . . . . . 10 (𝜑 → (𝑅‘(𝐺𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵)))
4342fveq2d 6911 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))))
4443adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘(𝐺𝐵))))
4539, 41, 443eqtr4d 2785 . . . . . . 7 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘suc (𝐺𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))))
461, 2, 3, 4, 5om2noseqrdg 28325 . . . . . . . . 9 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘(𝐺𝐵)) = ⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
4746fveq2d 6911 . . . . . . . 8 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩))
48 df-ov 7434 . . . . . . . 8 ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(𝐺𝐵)), (2nd ‘(𝑅‘(𝐺𝐵)))⟩)
4947, 48eqtr4di 2793 . . . . . . 7 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)‘(𝑅‘(𝐺𝐵))) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
5045, 49eqtrd 2775 . . . . . 6 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘suc (𝐺𝐵)) = ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))))
51 fvex 6920 . . . . . . 7 (𝐺‘(𝐺𝐵)) ∈ V
52 fvex 6920 . . . . . . 7 (2nd ‘(𝑅‘(𝐺𝐵))) ∈ V
53 oveq1 7438 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧 +s 1s ) = ((𝐺‘(𝐺𝐵)) +s 1s ))
54 oveq1 7438 . . . . . . . . 9 (𝑧 = (𝐺‘(𝐺𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹𝑤))
5553, 54opeq12d 4886 . . . . . . . 8 (𝑧 = (𝐺‘(𝐺𝐵)) → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩)
56 oveq2 7439 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ((𝐺‘(𝐺𝐵))𝐹𝑤) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
5756opeq2d 4885 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝐵))) → ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
58 oveq1 7438 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 +s 1s ) = (𝑧 +s 1s ))
59 oveq1 7438 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
6058, 59opeq12d 4886 . . . . . . . . 9 (𝑥 = 𝑧 → ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩)
61 oveq2 7439 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
6261opeq2d 4885 . . . . . . . . 9 (𝑦 = 𝑤 → ⟨(𝑧 +s 1s ), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
6360, 62cbvmpov 7528 . . . . . . . 8 (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 +s 1s ), (𝑧𝐹𝑤)⟩)
64 opex 5475 . . . . . . . 8 ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩ ∈ V
6555, 57, 63, 64ovmpo 7593 . . . . . . 7 (((𝐺‘(𝐺𝐵)) ∈ V ∧ (2nd ‘(𝑅‘(𝐺𝐵))) ∈ V) → ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
6651, 52, 65mp2an 692 . . . . . 6 ((𝐺‘(𝐺𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(𝐺𝐵)))) = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩
6750, 66eqtrdi 2791 . . . . 5 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (𝑅‘suc (𝐺𝐵)) = ⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩)
6867fveq2d 6911 . . . 4 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (2nd ‘(𝑅‘suc (𝐺𝐵))) = (2nd ‘⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩))
69 ovex 7464 . . . . 5 ((𝐺‘(𝐺𝐵)) +s 1s ) ∈ V
70 ovex 7464 . . . . 5 ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) ∈ V
7169, 70op2nd 8022 . . . 4 (2nd ‘⟨((𝐺‘(𝐺𝐵)) +s 1s ), ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))⟩) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵))))
7268, 71eqtrdi 2791 . . 3 ((𝜑 ∧ (𝐺𝐵) ∈ ω) → (2nd ‘(𝑅‘suc (𝐺𝐵))) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
7323, 72syldan 591 . 2 ((𝜑𝐵𝑍) → (2nd ‘(𝑅‘suc (𝐺𝐵))) = ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))))
741, 2, 3, 4, 5noseqrdglem 28326 . . . . . 6 ((𝜑𝐵𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
7574, 16eleqtrrd 2842 . . . . 5 ((𝜑𝐵𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆)
76 funopfv 6959 . . . . 5 (Fun 𝑆 → (⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ 𝑆 → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵)))))
779, 75, 76sylc 65 . . . 4 ((𝜑𝐵𝑍) → (𝑆𝐵) = (2nd ‘(𝑅‘(𝐺𝐵))))
7877eqcomd 2741 . . 3 ((𝜑𝐵𝑍) → (2nd ‘(𝑅‘(𝐺𝐵))) = (𝑆𝐵))
7930, 78oveq12d 7449 . 2 ((𝜑𝐵𝑍) → ((𝐺‘(𝐺𝐵))𝐹(2nd ‘(𝑅‘(𝐺𝐵)))) = (𝐵𝐹(𝑆𝐵)))
8037, 73, 793eqtrd 2779 1 ((𝜑𝐵𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cmpt 5231  ccnv 5688  ran crn 5690  cres 5691  cima 5692  suc csuc 6388  Fun wfun 6557   Fn wfn 6558  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  2nd c2nd 8012  reccrdg 8448   No csur 27699   1s c1s 27883   +s cadds 28007
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
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-ral 3060  df-rex 3069  df-rmo 3378  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-tp 4636  df-op 4638  df-ot 4640  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-se 5642  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-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008
This theorem is referenced by:  seqsp1  28332
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