Theorem nfseqs 28434

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 28431	. 2 ⊢ seqs𝑀( + , 𝐹) = (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
2		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 +s 1s )
4		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
5		nfseqs.2	. . . . . . 7 ⊢ Ⅎ𝑥 +
6		nfseqs.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
7	6, 3	nffv 6881	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑦 +s 1s ))
8	4, 5, 7	nfov 7430	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 + (𝐹‘(𝑦 +s 1s )))
9	3, 8	nfop 4849	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩
10	2, 2, 9	nfmpo 7482	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩)
11		nfseqs.1	. . . . 5 ⊢ Ⅎ𝑥𝑀
12	6, 11	nffv 6881	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀)
13	11, 12	nfop 4849	. . . 4 ⊢ Ⅎ𝑥⟨𝑀, (𝐹‘𝑀)⟩
14	10, 13	nfrdg 8389	. . 3 ⊢ Ⅎ𝑥rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15		nfcv 2927	. . 3 ⊢ Ⅎ𝑥ω
16	14, 15	nfima 6060	. 2 ⊢ Ⅎ𝑥(rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑧 + (𝐹‘(𝑦 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
17	1, 16	nfcxfr 2925	1 ⊢ Ⅎ𝑥seqs𝑀( + , 𝐹)

Syntax hints:  Ⅎwnfc 2912  Vcvv 3457  ⟨cop 4591   “ cima 5654  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ωcom 7850  reccrdg 8384   1_s c1s 27953   +_s cadds 28106  seq_scseqs 28430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-iota 6481  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-seqs 28431

This theorem is referenced by: (None)