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Theorem nfseq 12806
Description: Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfseq.1 𝑥𝑀
nfseq.2 𝑥 +
nfseq.3 𝑥𝐹
Assertion
Ref Expression
nfseq 𝑥seq𝑀( + , 𝐹)

Proof of Theorem nfseq
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-seq 12797 . 2 seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
2 nfcv 2763 . . . . 5 𝑥V
3 nfcv 2763 . . . . . 6 𝑥(𝑧 + 1)
4 nfcv 2763 . . . . . . 7 𝑥𝑤
5 nfseq.2 . . . . . . 7 𝑥 +
6 nfseq.3 . . . . . . . 8 𝑥𝐹
76, 3nffv 6196 . . . . . . 7 𝑥(𝐹‘(𝑧 + 1))
84, 5, 7nfov 6673 . . . . . 6 𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
93, 8nfop 4416 . . . . 5 𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
102, 2, 9nfmpt2 6721 . . . 4 𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
11 nfseq.1 . . . . 5 𝑥𝑀
126, 11nffv 6196 . . . . 5 𝑥(𝐹𝑀)
1311, 12nfop 4416 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1410, 13nfrdg 7507 . . 3 𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
15 nfcv 2763 . . 3 𝑥ω
1614, 15nfima 5472 . 2 𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
171, 16nfcxfr 2761 1 𝑥seq𝑀( + , 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2750  Vcvv 3198  cop 4181  cima 5115  cfv 5886  (class class class)co 6647  cmpt2 6649  ωcom 7062  reccrdg 7502  1c1 9934   + caddc 9936  seqcseq 12796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-iota 5849  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-seq 12797
This theorem is referenced by:  seqof2  12854  nfsum1  14414  nfsum  14415  nfcprod1  14634  nfcprod  14635  lgamgulm2  24756  binomcxplemdvbinom  38378  binomcxplemdvsum  38380  binomcxplemnotnn0  38381  fmuldfeqlem1  39620  fmuldfeq  39621  sumnnodd  39668  stoweidlem51  40037
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