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Theorem iseqvalcbv 9837
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 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑎 ∈ (ℤ𝑀), 𝑏𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹𝑀)⟩)
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
StepHypRef Expression
1 oveq1 5641 . . . . . . . . . 10 (𝑐 = 𝑧 → (𝑐 + 1) = (𝑧 + 1))
21fveq2d 5293 . . . . . . . . 9 (𝑐 = 𝑧 → (𝐹‘(𝑐 + 1)) = (𝐹‘(𝑧 + 1)))
32oveq2d 5650 . . . . . . . 8 (𝑐 = 𝑧 → (𝑑 + (𝐹‘(𝑐 + 1))) = (𝑑 + (𝐹‘(𝑧 + 1))))
4 oveq1 5641 . . . . . . . 8 (𝑑 = 𝑤 → (𝑑 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘(𝑧 + 1))))
53, 4cbvmpt2v 5710 . . . . . . 7 (𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1)))) = (𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
65oveqi 5647 . . . . . 6 (𝑥(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦) = (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)
76opeq2i 3621 . . . . 5 ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩
87a1i 9 . . . 4 ((𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑇) → ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
98mpt2eq3ia 5696 . . 3 (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
10 oveq1 5641 . . . . 5 (𝑥 = 𝑎 → (𝑥 + 1) = (𝑎 + 1))
11 oveq1 5641 . . . . 5 (𝑥 = 𝑎 → (𝑥(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦) = (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦))
1210, 11opeq12d 3625 . . . 4 (𝑥 = 𝑎 → ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩)
13 oveq2 5642 . . . . 5 (𝑦 = 𝑏 → (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦) = (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏))
1413opeq2d 3624 . . . 4 (𝑦 = 𝑏 → ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩)
1512, 14cbvmpt2v 5710 . . 3 (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩) = (𝑎 ∈ (ℤ𝑀), 𝑏𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩)
169, 15eqtr3i 2110 . 2 (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑎 ∈ (ℤ𝑀), 𝑏𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩)
17 freceq1 6139 . 2 ((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑎 ∈ (ℤ𝑀), 𝑏𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩) → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑎 ∈ (ℤ𝑀), 𝑏𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹𝑀)⟩))
1816, 17ax-mp 7 1 frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑎 ∈ (ℤ𝑀), 𝑏𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹𝑀)⟩)
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wcel 1438  cop 3444  cfv 5002  (class class class)co 5634  cmpt2 5636  freccfrec 6137  1c1 7330   + caddc 7332  cuz 8988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-res 4440  df-iota 4967  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-recs 6052  df-frec 6138
This theorem is referenced by:  iseq1t  9841  seq3-1  9842  iseqfclt  9844  seqf  9845  iseqp1t  9848  seq3p1  9849
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