Theorem seqeq2 13912

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.

Step	Hyp	Ref	Expression
1		oveq 7352	. . . . . 6 ⊢ ( + = 𝑄 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦𝑄(𝐹‘(𝑥 + 1))))
2	1	opeq2d 4829	. . . . 5 ⊢ ( + = 𝑄 → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩)
3	2	mpoeq3dv 7425	. . . 4 ⊢ ( + = 𝑄 → (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩))
4		rdgeq1 8330	. . . 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, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
5	3, 4	syl 17	. . 3 ⊢ ( + = 𝑄 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
6	5	imaeq1d 6007	. 2 ⊢ ( + = 𝑄 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω))
7		df-seq 13909	. 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
8		df-seq 13909	. 2 ⊢ seq𝑀(𝑄, 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
9	6, 7, 8	3eqtr4g 2791	1 ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Syntax hints:   → wi 4   = wceq 1541  Vcvv 3436  ⟨cop 4579   “ cima 5617  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  ωcom 7796  reccrdg 8328  1c1 11007   + caddc 11009  seqcseq 13908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-xp 5620  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-iota 6437  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-seq 13909

This theorem is referenced by:  seqeq2d  13915  sadcom  16374  ressmulgnn  18989  cvmliftlem15  35342