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Theorem om2noseqfo 28305
Description: Function statement for 𝐺. (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))
Assertion
Ref Expression
om2noseqfo (𝜑𝐺:ω–onto𝑍)

Proof of Theorem om2noseqfo
StepHypRef Expression
1 frfnom 8476 . . 3 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) Fn ω
2 om2noseq.2 . . . 4 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
32fneq1d 6660 . . 3 (𝜑 → (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) Fn ω))
41, 3mpbiri 258 . 2 (𝜑𝐺 Fn ω)
5 df-ima 5697 . . . 4 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
65eqcomi 2745 . . 3 ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)
72rneqd 5948 . . 3 (𝜑 → ran 𝐺 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
8 om2noseq.3 . . 3 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
96, 7, 83eqtr4a 2802 . 2 (𝜑 → ran 𝐺 = 𝑍)
10 df-fo 6566 . 2 (𝐺:ω–onto𝑍 ↔ (𝐺 Fn ω ∧ ran 𝐺 = 𝑍))
114, 9, 10sylanbrc 583 1 (𝜑𝐺:ω–onto𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  Vcvv 3479  cmpt 5224  ran crn 5685  cres 5686  cima 5687   Fn wfn 6555  ontowfo 6558  (class class class)co 7432  ωcom 7888  reccrdg 8450   No csur 27685   1s c1s 27869   +s cadds 27993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451
This theorem is referenced by:  om2noseqlt  28306  om2noseqlt2  28307  om2noseqf1o  28308  noseqrdgfn  28313
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