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Theorem noseqrdgfn 28327
Description: The recursive definition generator on surreal sequences is a function. (Contributed by Scott Fenton, 18-Apr-2025.)
Hypotheses
Ref Expression
om2noseq.1 (𝜑𝐶 No )
om2noseq.2 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseq.3 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
noseqrdg.1 (𝜑𝐴𝑉)
noseqrdg.2 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
noseqrdg.3 (𝜑𝑆 = ran 𝑅)
Assertion
Ref Expression
noseqrdgfn (𝜑𝑆 Fn 𝑍)
Distinct variable groups:   𝑥,𝐶   𝑥,𝐹,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem noseqrdgfn
Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noseqrdg.3 . . . . . . . 8 (𝜑𝑆 = ran 𝑅)
21eleq2d 2825 . . . . . . 7 (𝜑 → (𝑧𝑆𝑧 ∈ ran 𝑅))
3 frfnom 8474 . . . . . . . . 9 (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω
4 noseqrdg.2 . . . . . . . . . 10 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
54fneq1d 6662 . . . . . . . . 9 (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω))
63, 5mpbiri 258 . . . . . . . 8 (𝜑𝑅 Fn ω)
7 fvelrnb 6969 . . . . . . . 8 (𝑅 Fn ω → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = 𝑧))
86, 7syl 17 . . . . . . 7 (𝜑 → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = 𝑧))
92, 8bitrd 279 . . . . . 6 (𝜑 → (𝑧𝑆 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = 𝑧))
10 om2noseq.1 . . . . . . . . . 10 (𝜑𝐶 No )
11 om2noseq.2 . . . . . . . . . 10 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
12 om2noseq.3 . . . . . . . . . 10 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
13 noseqrdg.1 . . . . . . . . . 10 (𝜑𝐴𝑉)
1410, 11, 12, 13, 4om2noseqrdg 28325 . . . . . . . . 9 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
1510, 11, 12om2noseqfo 28319 . . . . . . . . . . . 12 (𝜑𝐺:ω–onto𝑍)
16 fof 6821 . . . . . . . . . . . 12 (𝐺:ω–onto𝑍𝐺:ω⟶𝑍)
1715, 16syl 17 . . . . . . . . . . 11 (𝜑𝐺:ω⟶𝑍)
1817ffvelcdmda 7104 . . . . . . . . . 10 ((𝜑𝑤 ∈ ω) → (𝐺𝑤) ∈ 𝑍)
19 fvex 6920 . . . . . . . . . 10 (2nd ‘(𝑅𝑤)) ∈ V
20 opelxpi 5726 . . . . . . . . . 10 (((𝐺𝑤) ∈ 𝑍 ∧ (2nd ‘(𝑅𝑤)) ∈ V) → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ (𝑍 × V))
2118, 19, 20sylancl 586 . . . . . . . . 9 ((𝜑𝑤 ∈ ω) → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ (𝑍 × V))
2214, 21eqeltrd 2839 . . . . . . . 8 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) ∈ (𝑍 × V))
23 eleq1 2827 . . . . . . . 8 ((𝑅𝑤) = 𝑧 → ((𝑅𝑤) ∈ (𝑍 × V) ↔ 𝑧 ∈ (𝑍 × V)))
2422, 23syl5ibcom 245 . . . . . . 7 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = 𝑧𝑧 ∈ (𝑍 × V)))
2524rexlimdva 3153 . . . . . 6 (𝜑 → (∃𝑤 ∈ ω (𝑅𝑤) = 𝑧𝑧 ∈ (𝑍 × V)))
269, 25sylbid 240 . . . . 5 (𝜑 → (𝑧𝑆𝑧 ∈ (𝑍 × V)))
2726ssrdv 4001 . . . 4 (𝜑𝑆 ⊆ (𝑍 × V))
28 relxp 5707 . . . 4 Rel (𝑍 × V)
29 relss 5794 . . . 4 (𝑆 ⊆ (𝑍 × V) → (Rel (𝑍 × V) → Rel 𝑆))
3027, 28, 29mpisyl 21 . . 3 (𝜑 → Rel 𝑆)
311eleq2d 2825 . . . . . . . 8 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅))
32 fvelrnb 6969 . . . . . . . . 9 (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩))
336, 32syl 17 . . . . . . . 8 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩))
3431, 33bitrd 279 . . . . . . 7 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩))
3514eqeq1d 2737 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
3635biimpd 229 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
3736impr 454 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩)) → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩)
38 fvex 6920 . . . . . . . . . . . . . 14 (𝐺𝑤) ∈ V
3938, 19opth1 5486 . . . . . . . . . . . . 13 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣)
4037, 39syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩)) → (𝐺𝑤) = 𝑣)
4110, 11, 12om2noseqf1o 28322 . . . . . . . . . . . . . 14 (𝜑𝐺:ω–1-1-onto𝑍)
42 f1ocnvfv 7298 . . . . . . . . . . . . . 14 ((𝐺:ω–1-1-onto𝑍𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4341, 42sylan 580 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4443adantrr 717 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩)) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4540, 44mpd 15 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩)) → (𝐺𝑣) = 𝑤)
4645fveq2d 6911 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩)) → (𝑅‘(𝐺𝑣)) = (𝑅𝑤))
4746fveq2d 6911 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩)) → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
48 vex 3482 . . . . . . . . . . 11 𝑣 ∈ V
49 vex 3482 . . . . . . . . . . 11 𝑧 ∈ V
5048, 49op2ndd 8024 . . . . . . . . . 10 ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅𝑤)) = 𝑧)
5150ad2antll 729 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩)) → (2nd ‘(𝑅𝑤)) = 𝑧)
5247, 51eqtr2d 2776 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩)) → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5352rexlimdvaa 3154 . . . . . . 7 (𝜑 → (∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5434, 53sylbid 240 . . . . . 6 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5554alrimiv 1925 . . . . 5 (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
56 fvex 6920 . . . . . 6 (2nd ‘(𝑅‘(𝐺𝑣))) ∈ V
57 eqeq2 2747 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (𝑧 = 𝑤𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5857imbi2d 340 . . . . . . 7 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
5958albidv 1918 . . . . . 6 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
6056, 59spcev 3606 . . . . 5 (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤))
6155, 60syl 17 . . . 4 (𝜑 → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤))
6261alrimiv 1925 . . 3 (𝜑 → ∀𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤))
63 dffun5 6580 . . 3 (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)))
6430, 62, 63sylanbrc 583 . 2 (𝜑 → Fun 𝑆)
65 dmss 5916 . . . . 5 (𝑆 ⊆ (𝑍 × V) → dom 𝑆 ⊆ dom (𝑍 × V))
6627, 65syl 17 . . . 4 (𝜑 → dom 𝑆 ⊆ dom (𝑍 × V))
67 dmxpss 6193 . . . 4 dom (𝑍 × V) ⊆ 𝑍
6866, 67sstrdi 4008 . . 3 (𝜑 → dom 𝑆𝑍)
6910, 11, 12, 13, 4noseqrdglem 28326 . . . . 5 ((𝜑𝑣𝑍) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ ran 𝑅)
701adantr 480 . . . . 5 ((𝜑𝑣𝑍) → 𝑆 = ran 𝑅)
7169, 70eleqtrrd 2842 . . . 4 ((𝜑𝑣𝑍) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆)
7248, 56opeldm 5921 . . . 4 (⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆𝑣 ∈ dom 𝑆)
7371, 72syl 17 . . 3 ((𝜑𝑣𝑍) → 𝑣 ∈ dom 𝑆)
7468, 73eqelssd 4017 . 2 (𝜑 → dom 𝑆 = 𝑍)
75 df-fn 6566 . 2 (𝑆 Fn 𝑍 ↔ (Fun 𝑆 ∧ dom 𝑆 = 𝑍))
7664, 74, 75sylanbrc 583 1 (𝜑𝑆 Fn 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  wrex 3068  Vcvv 3478  wss 3963  cop 4637  cmpt 5231   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Rel wrel 5694  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  2nd c2nd 8012  reccrdg 8448   No csur 27699   1s c1s 27883   +s cadds 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008
This theorem is referenced by:  noseqrdg0  28328  noseqrdgsuc  28329  seqsfn  28330
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