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Theorem iseqeq4 9217
 Description: Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
iseqeq4 (𝑆 = 𝑇 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))

Proof of Theorem iseqeq4
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2040 . . . . 5 (ℤ𝑀) = (ℤ𝑀)
2 mpt2eq12 5565 . . . . 5 (((ℤ𝑀) = (ℤ𝑀) ∧ 𝑆 = 𝑇) → (𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩))
31, 2mpan 400 . . . 4 (𝑆 = 𝑇 → (𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩))
4 freceq1 5979 . . . 4 ((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
53, 4syl 14 . . 3 (𝑆 = 𝑇 → frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
65rneqd 4563 . 2 (𝑆 = 𝑇 → ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
7 df-iseq 9212 . 2 seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
8 df-iseq 9212 . 2 seq𝑀( + , 𝐹, 𝑇) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
96, 7, 83eqtr4g 2097 1 (𝑆 = 𝑇 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  ⟨cop 3378  ran crn 4346  ‘cfv 4902  (class class class)co 5512   ↦ cmpt2 5514  freccfrec 5977  1c1 6890   + caddc 6892  ℤ≥cuz 8473  seqcseq 9211 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-frec 5978  df-iseq 9212 This theorem is referenced by: (None)
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