Theorem seqval 12674

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 5051	. 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 12664	. 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 2610	. . . . . . 7 ⊢ V = V
5		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
6		vex 3176	. . . . . . . . 9 ⊢ 𝑦 ∈ V
7		oveq1 6556	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 + 1) = (𝑥 + 1))
8	7	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝐹‘(𝑧 + 1)) = (𝐹‘(𝑥 + 1)))
9	8	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘(𝑥 + 1))))
10		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘(𝑥 + 1))))
11		eqid 2610	. . . . . . . . . 10 ⊢ (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
12		ovex 6577	. . . . . . . . . 10 ⊢ (𝑦 + (𝐹‘(𝑥 + 1))) ∈ V
13	9, 10, 11, 12	ovmpt2 6694	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = (𝑦 + (𝐹‘(𝑥 + 1))))
14	5, 6, 13	mp2an 704	. . . . . . . 8 ⊢ (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = (𝑦 + (𝐹‘(𝑥 + 1)))
15	14	opeq2i 4344	. . . . . . 7 ⊢ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
16	4, 4, 15	mpt2eq123i 6616	. . . . . 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)))⟩)
17		rdgeq1 7394	. . . . . 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)⟩))
18	16, 17	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)⟩)
19	18	reseq1i 5313	. . . 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)
20	3, 19	eqtri 2632	. . 3 ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)
21	20	rneqi 5273	. 2 ⊢ ran 𝑅 = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)
22	1, 2, 21	3eqtr4i 2642	1 ⊢ seq𝑀( + , 𝐹) = ran 𝑅

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ran crn 5039   ↾ cres 5040   “ cima 5041  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  reccrdg 7392  1c1 9816   + caddc 9818  seqcseq 12663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664

This theorem is referenced by:  seqfn  12675  seq1  12676  seqp1  12678