Theorem nfseqs 28293

Description: Hypothesis builder for the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypotheses

Ref	Expression
nfseqs.1	⊢ Ⅎ𝑥𝑀
nfseqs.2	⊢ Ⅎ𝑥 +
nfseqs.3	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
nfseqs	⊢ Ⅎ𝑥seqs𝑀( + , 𝐹)

Proof of Theorem nfseqs

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-seqs 28290	. 2 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
2		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 +s 1s )
4		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
5		nfseqs.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
6		nfseqs.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
7	6, 3	nffv 6844	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑦 +s 1s ))
8	4, 5, 7	nfov 7390	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + (𝐹‘(𝑦 +s 1s )))
9	3, 8	nfop 4833	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩
10	2, 2, 9	nfmpo 7442	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩)
11		nfseqs.1	. . . . 5 ⊢ Ⅎ𝑥𝑀
12	6, 11	nffv 6844	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
13	11, 12	nfop 4833	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
14	10, 13	nfrdg 8346	. . 3 ⊢ Ⅎ𝑥rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15		nfcv 2899	. . 3 ⊢ Ⅎ𝑥ω
16	14, 15	nfima 6027	. 2 ⊢ Ⅎ𝑥(rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
17	1, 16	nfcxfr 2897	1 ⊢ Ⅎ𝑥seqs𝑀( + , 𝐹)

Syntax hints:  Ⅎwnfc 2884  Vcvv 3430  ⟨cop 4574   “ cima 5627  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  ωcom 7810  reccrdg 8341   1_s c1s 27812   +_s cadds 27965  seq_scseqs 28289

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5630  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-seqs 28290

This theorem is referenced by: (None)