Theorem enumctlemm 6992

Description: Lemma for enumct 6993. The case where 𝑁 is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)

Hypotheses

Ref	Expression
enumctlemm.f	⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)
enumctlemm.n	⊢ (𝜑 → 𝑁 ∈ ω)
enumctlemm.n0	⊢ (𝜑 → ∅ ∈ 𝑁)
enumctlemm.g	⊢ 𝐺 = (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑁, (𝐹‘𝑘), (𝐹‘∅)))

Assertion

Ref	Expression
enumctlemm	⊢ (𝜑 → 𝐺:ω–onto→𝐴)

Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑁   𝜑,𝑘

Allowed substitution hint:   𝐺(𝑘)

Proof of Theorem enumctlemm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enumctlemm.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)
2		fof 5340	. . . . . . 7 ⊢ (𝐹:𝑁–onto→𝐴 → 𝐹:𝑁⟶𝐴)
3	1, 2	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑁⟶𝐴)
4	3	ffvelrnda 5548	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → (𝐹‘𝑘) ∈ 𝐴)
5	4	adantlr 468	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ 𝑁) → (𝐹‘𝑘) ∈ 𝐴)
6		enumctlemm.n0	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑁)
7	3, 6	ffvelrnd 5549	. . . . 5 ⊢ (𝜑 → (𝐹‘∅) ∈ 𝐴)
8	7	ad2antrr 479	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ ω) ∧ ¬ 𝑘 ∈ 𝑁) → (𝐹‘∅) ∈ 𝐴)
9		simpr 109	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → 𝑘 ∈ ω)
10		enumctlemm.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
11	10	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → 𝑁 ∈ ω)
12		nndcel 6389	. . . . 5 ⊢ ((𝑘 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑘 ∈ 𝑁)
13	9, 11, 12	syl2anc 408	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → DECID 𝑘 ∈ 𝑁)
14	5, 8, 13	ifcldadc 3496	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → if(𝑘 ∈ 𝑁, (𝐹‘𝑘), (𝐹‘∅)) ∈ 𝐴)
15		enumctlemm.g	. . 3 ⊢ 𝐺 = (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑁, (𝐹‘𝑘), (𝐹‘∅)))
16	14, 15	fmptd 5567	. 2 ⊢ (𝜑 → 𝐺:ω⟶𝐴)
17		foelrn 5647	. . . . . 6 ⊢ ((𝐹:𝑁–onto→𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝑁 𝑦 = (𝐹‘𝑥))
18	1, 17	sylan 281	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝑁 𝑦 = (𝐹‘𝑥))
19		eleq1w 2198	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (𝑘 ∈ 𝑁 ↔ 𝑥 ∈ 𝑁))
20		fveq2 5414	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
21	19, 20	ifbieq1d 3489	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → if(𝑘 ∈ 𝑁, (𝐹‘𝑘), (𝐹‘∅)) = if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)))
22		simpr 109	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ 𝑁)
23	10	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑁 ∈ ω)
24		elnn 4514	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑁 ∈ ω) → 𝑥 ∈ ω)
25	22, 23, 24	syl2anc 408	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ ω)
26	22	iftrued 3476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)) = (𝐹‘𝑥))
27	3	ffvelrnda 5548	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐹‘𝑥) ∈ 𝐴)
28	26, 27	eqeltrd 2214	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)) ∈ 𝐴)
29	15, 21, 25, 28	fvmptd3 5507	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐺‘𝑥) = if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)))
30	29, 26	eqtrd 2170	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐺‘𝑥) = (𝐹‘𝑥))
31	30	eqeq2d 2149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝑦 = (𝐺‘𝑥) ↔ 𝑦 = (𝐹‘𝑥)))
32	31	rexbidva 2432	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ 𝑁 𝑦 = (𝐺‘𝑥) ↔ ∃𝑥 ∈ 𝑁 𝑦 = (𝐹‘𝑥)))
33	32	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑥 ∈ 𝑁 𝑦 = (𝐺‘𝑥) ↔ ∃𝑥 ∈ 𝑁 𝑦 = (𝐹‘𝑥)))
34	18, 33	mpbird 166	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝑁 𝑦 = (𝐺‘𝑥))
35		omelon 4517	. . . . . . 7 ⊢ ω ∈ On
36	35	onelssi 4346	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ⊆ ω)
37		ssrexv 3157	. . . . . 6 ⊢ (𝑁 ⊆ ω → (∃𝑥 ∈ 𝑁 𝑦 = (𝐺‘𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥)))
38	10, 36, 37	3syl 17	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑁 𝑦 = (𝐺‘𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥)))
39	38	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑥 ∈ 𝑁 𝑦 = (𝐺‘𝑥) → ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥)))
40	34, 39	mpd 13	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥))
41	40	ralrimiva 2503	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥))
42		dffo3 5560	. 2 ⊢ (𝐺:ω–onto→𝐴 ↔ (𝐺:ω⟶𝐴 ∧ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥)))
43	16, 41, 42	sylanbrc 413	1 ⊢ (𝜑 → 𝐺:ω–onto→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 819   = wceq 1331   ∈ wcel 1480  ∀wral 2414  ∃wrex 2415   ⊆ wss 3066  ∅c0 3358  ifcif 3469   ↦ cmpt 3984  ωcom 4499  ⟶wf 5114  –onto→wfo 5116  ‘cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126

This theorem is referenced by:  enumct  6993