Theorem om2noseqfo 28319

Description: Function statement for 𝐺. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypotheses

Ref	Expression
om2noseq.1	⊢ (𝜑 → 𝐶 ∈ No )
om2noseq.2	⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseq.3	⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))

Assertion

Ref	Expression
om2noseqfo	⊢ (𝜑 → 𝐺:ω–onto→𝑍)

Proof of Theorem om2noseqfo

Step	Hyp	Ref	Expression
1		frfnom 8474	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) Fn ω
2		om2noseq.2	. . . 4 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
3	2	fneq1d 6662	. . 3 ⊢ (𝜑 → (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) Fn ω))
4	1, 3	mpbiri 258	. 2 ⊢ (𝜑 → 𝐺 Fn ω)
5		df-ima 5702	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
6	5	eqcomi 2744	. . 3 ⊢ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)
7	2	rneqd 5952	. . 3 ⊢ (𝜑 → ran 𝐺 = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
8		om2noseq.3	. . 3 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
9	6, 7, 8	3eqtr4a 2801	. 2 ⊢ (𝜑 → ran 𝐺 = 𝑍)
10		df-fo 6569	. 2 ⊢ (𝐺:ω–onto→𝑍 ↔ (𝐺 Fn ω ∧ ran 𝐺 = 𝑍))
11	4, 9, 10	sylanbrc 583	1 ⊢ (𝜑 → 𝐺:ω–onto→𝑍)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ↦ cmpt 5231  ran crn 5690   ↾ cres 5691   “ cima 5692   Fn wfn 6558  –onto→wfo 6561  (class class class)co 7431  ωcom 7887  reccrdg 8448   No csur 27699   1_s 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-sep 5302  ax-nul 5312  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-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-op 4638  df-uni 4913  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-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-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449

This theorem is referenced by:  om2noseqlt  28320  om2noseqlt2  28321  om2noseqf1o  28322  noseqrdgfn  28327