Theorem seqsfn 28710

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 2764	. 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 7437	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
5	4	cbvmptv 5212	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑥 +s 1s )) = (𝑦 ∈ V ↦ (𝑦 +s 1s ))
6		rdgeq1 8433	. . . . 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 6055	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω) = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω)
9	3, 8	eqtrdi 2814	. 2 ⊢ (𝜑 → 𝑍 = (rec((𝑦 ∈ V ↦ (𝑦 +s 1s )), 𝑀) “ ω))
10		fvexd 6911	. 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ V)
11		eqidd 2764	. 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 28689	. 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 28707	1 ⊢ (𝜑 → seqs𝑀( + , 𝐹) Fn 𝑍)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  Vcvv 3454  ⟨cop 4838   ↦ cmpt 5189   ↾ cres 5657   “ cima 5658   Fn wfn 6542  ‘cfv 6547  (class class class)co 7430   ∈ cmpo 7432  ωcom 7887  reccrdg 8431  Nocsur 28012   1_s c1s 28207   +_s cadds 28360  seq_scseqs 28684

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-2o 8491  df-oadd 8494  df-nadd 8689  df-no 28015  df-lts 28016  df-bday 28017  df-les 28117  df-slts 28159  df-cuts 28161  df-0s 28208  df-1s 28209  df-made 28228  df-old 28229  df-left 28231  df-right 28232  df-norec2 28350  df-adds 28361  df-seqs 28685

This theorem is used by:  seqn0sfn  28761