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Theorem seqeq2 13368
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq2 ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Proof of Theorem seqeq2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq 7141 . . . . . 6 ( + = 𝑄 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦𝑄(𝐹‘(𝑥 + 1))))
21opeq2d 4772 . . . . 5 ( + = 𝑄 → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩)
32mpoeq3dv 7212 . . . 4 ( + = 𝑄 → (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩))
4 rdgeq1 8030 . . . 4 ((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
53, 4syl 17 . . 3 ( + = 𝑄 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩))
65imaeq1d 5895 . 2 ( + = 𝑄 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω))
7 df-seq 13365 . 2 seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
8 df-seq 13365 . 2 seq𝑀(𝑄, 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
96, 7, 83eqtr4g 2858 1 ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  Vcvv 3441  cop 4531  cima 5522  cfv 6324  (class class class)co 7135  cmpo 7137  ωcom 7560  reccrdg 8028  1c1 10527   + caddc 10529  seqcseq 13364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seq 13365
This theorem is referenced by:  seqeq2d  13371  sadcom  15802  ressmulgnn  30717  cvmliftlem15  32655
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