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Theorem nfseqs 28308
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 28305 . 2 seqs𝑀( + , 𝐹) = (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
2 nfcv 2903 . . . . 5 𝑥V
3 nfcv 2903 . . . . . 6 𝑥(𝑦 +s 1s )
4 nfcv 2903 . . . . . . 7 𝑥𝑧
5 nfseqs.2 . . . . . . 7 𝑥 +
6 nfseqs.3 . . . . . . . 8 𝑥𝐹
76, 3nffv 6917 . . . . . . 7 𝑥(𝐹‘(𝑦 +s 1s ))
84, 5, 7nfov 7461 . . . . . 6 𝑥(𝑧 + (𝐹‘(𝑦 +s 1s )))
93, 8nfop 4894 . . . . 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 6917 . . . . 5 𝑥(𝐹𝑀)
1311, 12nfop 4894 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1410, 13nfrdg 8453 . . 3 𝑥rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩)
15 nfcv 2903 . . 3 𝑥ω
1614, 15nfima 6088 . 2 𝑥(rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
171, 16nfcxfr 2901 1 𝑥seqs𝑀( + , 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2888  Vcvv 3478  cop 4637  cima 5692  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  reccrdg 8448   1s c1s 27883   +s cadds 28007  seqscseqs 28304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqs 28305
This theorem is referenced by: (None)
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