Theorem om2noseqfo 28197

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 8357	. . 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 6575	. . 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 5632	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
6	5	eqcomi 2738	. . 3 ⊢ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)
7	2	rneqd 5880	. . 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 2790	. 2 ⊢ (𝜑 → ran 𝐺 = 𝑍)
10		df-fo 6488	. 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 1540   ∈ wcel 2109  Vcvv 3436   ↦ cmpt 5173  ran crn 5620   ↾ cres 5621   “ cima 5622   Fn wfn 6477  –onto→wfo 6480  (class class class)co 7349  ωcom 7799  reccrdg 8331   No csur 27549   1_s c1s 27737   +_s cadds 27871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332

This theorem is referenced by:  om2noseqlt  28198  om2noseqlt2  28199  om2noseqf1o  28200  noseqrdgfn  28205