Theorem iseqvalcbv 10845

Description: Changing the bound variables in an expression which appears in some seq related proofs. (Contributed by Jim Kingdon, 28-Apr-2022.)

Assertion

Ref	Expression
iseqvalcbv	⊢ frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)

Distinct variable groups:   + ,𝑎,𝑏,𝑐,𝑑,𝑥,𝑦   𝑤, + ,𝑧,𝑐,𝑑   𝐹,𝑎,𝑏,𝑐,𝑑,𝑥,𝑦   𝑤,𝐹,𝑧   𝑀,𝑎,𝑏,𝑐,𝑑,𝑥,𝑦   𝑤,𝑀,𝑧   𝑆,𝑎,𝑏,𝑐,𝑑,𝑥,𝑦   𝑤,𝑆,𝑧   𝑇,𝑎,𝑏,𝑥,𝑦

Allowed substitution hints:   𝑇(𝑧,𝑤,𝑐,𝑑)

Proof of Theorem iseqvalcbv

Step	Hyp	Ref	Expression
1		oveq1 6065	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑐 + 1) = (𝑧 + 1))
2	1	fveq2d 5679	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝐹‘(𝑐 + 1)) = (𝐹‘(𝑧 + 1)))
3	2	oveq2d 6074	. . . . . . . 8 ⊢ (𝑐 = 𝑧 → (𝑑 + (𝐹‘(𝑐 + 1))) = (𝑑 + (𝐹‘(𝑧 + 1))))
4		oveq1 6065	. . . . . . . 8 ⊢ (𝑑 = 𝑤 → (𝑑 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘(𝑧 + 1))))
5	3, 4	cbvmpov 6141	. . . . . . 7 ⊢ (𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1)))) = (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
6	5	oveqi 6071	. . . . . 6 ⊢ (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦) = (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)
7	6	opeq2i 3892	. . . . 5 ⊢ ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩
8	7	a1i 9	. . . 4 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑇) → ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
9	8	mpoeq3ia 6126	. . 3 ⊢ (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
10		oveq1 6065	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 + 1) = (𝑎 + 1))
11		oveq1 6065	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦) = (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦))
12	10, 11	opeq12d 3896	. . . 4 ⊢ (𝑥 = 𝑎 → ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩)
13		oveq2 6066	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦) = (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏))
14	13	opeq2d 3895	. . . 4 ⊢ (𝑦 = 𝑏 → ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩)
15	12, 14	cbvmpov 6141	. . 3 ⊢ (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩) = (𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩)
16	9, 15	eqtr3i 2257	. 2 ⊢ (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩)
17		freceq1 6636	. 2 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
18	16, 17	ax-mp 5	1 ⊢ frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ⟨cop 3697  ‘cfv 5357  (class class class)co 6058   ∈ cmpo 6060  freccfrec 6634  1c1 8144   + caddc 8146  ℤ_≥cuz 9871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-res 4766  df-iota 5317  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-recs 6549  df-frec 6635

This theorem is referenced by:  seq3-1  10848  seqf  10850  seq3p1  10851  seqf2  10854  seq1cd  10855  seqp1cd  10856