Theorem om2noseqfo 28563

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 8424	. . 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 6625	. . 3 ⊢ (𝜑 → (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) Fn ω))
4	1, 3	mpbiri 261	. 2 ⊢ (𝜑 → 𝐺 Fn ω)
5		df-ima 5668	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)
6	5	eqcomi 2769	. . 3 ⊢ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)
7	2	rneqd 5922	. . 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 2821	. 2 ⊢ (𝜑 → ran 𝐺 = 𝑍)
10		df-fo 6539	. 2 ⊢ (𝐺:ω–onto→𝑍 ↔ (𝐺 Fn ω ∧ ran 𝐺 = 𝑍))
11	4, 9, 10	sylanbrc 595	1 ⊢ (𝜑 → 𝐺:ω–onto→𝑍)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ↦ cmpt 5186  ran crn 5656   ↾ cres 5657   “ cima 5658   Fn wfn 6528  –onto→wfo 6531  (class class class)co 7413  ωcom 7862  reccrdg 8398   No csur 27876   1_s c1s 28071   +_s cadds 28224

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399

This theorem is used by:  om2noseqlt  28564  om2noseqlt2  28565  om2noseqf1o  28566  noseqrdgfn  28571