Theorem iseqvalcbv 10392

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 5849	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑐 + 1) = (𝑧 + 1))
2	1	fveq2d 5490	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝐹‘(𝑐 + 1)) = (𝐹‘(𝑧 + 1)))
3	2	oveq2d 5858	. . . . . . . 8 ⊢ (𝑐 = 𝑧 → (𝑑 + (𝐹‘(𝑐 + 1))) = (𝑑 + (𝐹‘(𝑧 + 1))))
4		oveq1 5849	. . . . . . . 8 ⊢ (𝑑 = 𝑤 → (𝑑 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘(𝑧 + 1))))
5	3, 4	cbvmpov 5922	. . . . . . 7 ⊢ (𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1)))) = (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
6	5	oveqi 5855	. . . . . 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 3762	. . . . 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 5907	. . 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 5849	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 + 1) = (𝑎 + 1))
11		oveq1 5849	. . . . 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 3766	. . . 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 5850	. . . . 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 3765	. . . 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 5922	. . 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 2188	. 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 6360	. 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 103   = wceq 1343   ∈ wcel 2136  ⟨cop 3579  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844  freccfrec 6358  1c1 7754   + caddc 7756  ℤ_≥cuz 9466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-res 4616  df-iota 5153  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-recs 6273  df-frec 6359

This theorem is referenced by:  seq3-1  10395  seqf  10396  seq3p1  10397  seqf2  10399  seq1cd  10400  seqp1cd  10401