Theorem om2noseqfo 28294

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 8366	. . 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 6585	. . 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 5637	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
6	5	eqcomi 2745	. . 3 ⊢ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)
7	2	rneqd 5887	. . 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 2797	. 2 ⊢ (𝜑 → ran 𝐺 = 𝑍)
10		df-fo 6498	. 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 1541   ∈ wcel 2113  Vcvv 3440   ↦ cmpt 5179  ran crn 5625   ↾ cres 5626   “ cima 5627   Fn wfn 6487  –onto→wfo 6490  (class class class)co 7358  ωcom 7808  reccrdg 8340   No csur 27607   1_s c1s 27802   +_s cadds 27955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  om2noseqlt  28295  om2noseqlt2  28296  om2noseqf1o  28297  noseqrdgfn  28302