Theorem nfseqs 28210

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 28207	. 2 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
2		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 +s 1s )
4		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
5		nfseqs.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
6		nfseqs.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
7	6, 3	nffv 6906	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑦 +s 1s ))
8	4, 5, 7	nfov 7449	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + (𝐹‘(𝑦 +s 1s )))
9	3, 8	nfop 4891	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩
10	2, 2, 9	nfmpo 7502	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩)
11		nfseqs.1	. . . . 5 ⊢ Ⅎ𝑥𝑀
12	6, 11	nffv 6906	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
13	11, 12	nfop 4891	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
14	10, 13	nfrdg 8435	. . 3 ⊢ Ⅎ𝑥rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15		nfcv 2891	. . 3 ⊢ Ⅎ𝑥ω
16	14, 15	nfima 6072	. 2 ⊢ Ⅎ𝑥(rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
17	1, 16	nfcxfr 2889	1 ⊢ Ⅎ𝑥seqs𝑀( + , 𝐹)

Syntax hints:  Ⅎwnfc 2875  Vcvv 3461  ⟨cop 4636   “ cima 5681  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421  ωcom 7871  reccrdg 8430   1_s c1s 27802   +_s cadds 27922  seq_scseqs 28206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5684  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-iota 6501  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-seqs 28207

This theorem is referenced by: (None)