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Theorem nfseqs 28293
Description: Hypothesis builder for the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)
Hypotheses
Ref Expression
nfseqs.1 𝑥𝑀
nfseqs.2 𝑥 +
nfseqs.3 𝑥𝐹
Assertion
Ref Expression
nfseqs 𝑥seqs𝑀( + , 𝐹)

Proof of Theorem nfseqs
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-seqs 28290 . 2 seqs𝑀( + , 𝐹) = (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
2 nfcv 2905 . . . . 5 𝑥V
3 nfcv 2905 . . . . . 6 𝑥(𝑦 +s 1s )
4 nfcv 2905 . . . . . . 7 𝑥𝑧
5 nfseqs.2 . . . . . . 7 𝑥 +
6 nfseqs.3 . . . . . . . 8 𝑥𝐹
76, 3nffv 6916 . . . . . . 7 𝑥(𝐹‘(𝑦 +s 1s ))
84, 5, 7nfov 7461 . . . . . 6 𝑥(𝑧 + (𝐹‘(𝑦 +s 1s )))
93, 8nfop 4889 . . . . 5 𝑥⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩
102, 2, 9nfmpo 7515 . . . 4 𝑥(𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩)
11 nfseqs.1 . . . . 5 𝑥𝑀
126, 11nffv 6916 . . . . 5 𝑥(𝐹𝑀)
1311, 12nfop 4889 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1410, 13nfrdg 8454 . . 3 𝑥rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩)
15 nfcv 2905 . . 3 𝑥ω
1614, 15nfima 6086 . 2 𝑥(rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
171, 16nfcxfr 2903 1 𝑥seqs𝑀( + , 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2890  Vcvv 3480  cop 4632  cima 5688  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  reccrdg 8449   1s c1s 27868   +s cadds 27992  seqscseqs 28289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqs 28290
This theorem is referenced by: (None)
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