Theorem seqval 13916

Description: Value of the sequence builder function. (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
seqval.1	⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)

Assertion

Ref	Expression
seqval	⊢ seq𝑀( + , 𝐹) = ran 𝑅

Distinct variable groups:   𝑤,𝐹,𝑥,𝑦,𝑧   𝑤, + ,𝑥,𝑦,𝑧   𝑥,𝑀,𝑦

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧,𝑤)   𝑀(𝑧,𝑤)

Proof of Theorem seqval

Step	Hyp	Ref	Expression
1		df-ima 5629	. 2 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)
2		df-seq 13906	. 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
3		seqval.1	. . . 4 ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)
4		eqid 2731	. . . . . . 7 ⊢ V = V
5		fvoveq1 7369	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝐹‘(𝑧 + 1)) = (𝐹‘(𝑥 + 1)))
6	5	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘(𝑥 + 1))))
7		oveq1 7353	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘(𝑥 + 1))))
8		eqid 2731	. . . . . . . . . 10 ⊢ (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
9		ovex 7379	. . . . . . . . . 10 ⊢ (𝑦 + (𝐹‘(𝑥 + 1))) ∈ V
10	6, 7, 8, 9	ovmpo 7506	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = (𝑦 + (𝐹‘(𝑥 + 1))))
11	10	el2v 3443	. . . . . . . 8 ⊢ (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = (𝑦 + (𝐹‘(𝑥 + 1)))
12	11	opeq2i 4829	. . . . . . 7 ⊢ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
13	4, 4, 12	mpoeq123i 7422	. . . . . 6 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
14		rdgeq1 8330	. . . . . 6 ⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
15	13, 14	ax-mp 5	. . . . 5 ⊢ rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
16	15	reseq1i 5924	. . . 4 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)
17	3, 16	eqtri 2754	. . 3 ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)
18	17	rneqi 5877	. 2 ⊢ ran 𝑅 = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)
19	1, 2, 18	3eqtr4i 2764	1 ⊢ seq𝑀( + , 𝐹) = ran 𝑅

Syntax hints:   = wceq 1541  Vcvv 3436  ⟨cop 4582  ran crn 5617   ↾ cres 5618   “ cima 5619  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  ωcom 7796  reccrdg 8328  1c1 11004   + caddc 11006  seqcseq 13905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-seq 13906

This theorem is referenced by:  seqfn  13917  seq1  13918  seqp1  13920