Theorem seqsfn 28577

Description: The surreal sequence builder is a function. (Contributed by Scott Fenton, 19-Apr-2025.)

Hypotheses

Ref	Expression
seqsfn.1	⊢ (𝜑 → 𝑀 ∈ No )
seqsfn.2	⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))

Assertion

Ref	Expression
seqsfn	⊢ (𝜑 → seqs𝑀( + , 𝐹) Fn 𝑍)

Proof of Theorem seqsfn

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

Step	Hyp	Ref	Expression
1		seqsfn.1	. 2 ⊢ (𝜑 → 𝑀 ∈ No )
2		eqidd 2761	. 2 ⊢ (𝜑 → (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) ↾ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) ↾ ω))
3		seqsfn.2	. . 3 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))
4		oveq1 7421	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
5	4	cbvmptv 5209	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑥 +s 1s )) = (𝑦 ∈ V ↦ (𝑦 +s 1s ))
6		rdgeq1 8401	. . . . 5 ⊢ ((𝑥 ∈ V ↦ (𝑥 +s 1s )) = (𝑦 ∈ V ↦ (𝑦 +s 1s )) → rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) = rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀))
7	5, 6	ax-mp 5	. . . 4 ⊢ rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) = rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀)
8	7	imaeq1i 6053	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω)
9	3, 8	eqtrdi 2811	. 2 ⊢ (𝜑 → 𝑍 = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω))
10		fvexd 6894	. 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ V)
11		eqidd 2761	. 2 ⊢ (𝜑 → (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω) = (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω))
12	11	seqsval 28556	. 2 ⊢ (𝜑 → seqs𝑀( + , 𝐹) = ran (rec((𝑦 ∈ V, 𝑧 ∈ V ↦ ⟨(𝑦 +s 1s ), (𝑦(𝑤 ∈ V, 𝑡 ∈ V ↦ (𝑡 + (𝐹‘(𝑤 +s 1s ))))𝑧)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω))
13	1, 2, 9, 10, 11, 12	noseqrdgfn 28574	1 ⊢ (𝜑 → seqs𝑀( + , 𝐹) Fn 𝑍)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⟨cop 4590   ↦ cmpt 5186   ↾ cres 5657   “ cima 5658   Fn wfn 6528  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  ωcom 7863  reccrdg 8399   No csur 27879   1_s c1s 28074   +_s cadds 28227  seq_scseqs 28551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-2o 8459  df-oadd 8462  df-nadd 8657  df-no 27882  df-lts 27883  df-bday 27884  df-les 27984  df-slts 28026  df-cuts 28028  df-0s 28075  df-1s 28076  df-made 28095  df-old 28096  df-left 28098  df-right 28099  df-norec2 28217  df-adds 28228  df-seqs 28552

This theorem is used by:  seqn0sfn  28628