Theorem enumctlemm 7079

Description: Lemma for enumct 7080. 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 5410	. . . . . . 7 ⊢ (𝐹:𝑁–onto→𝐴 → 𝐹:𝑁⟶𝐴)
3	1, 2	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑁⟶𝐴)
4	3	ffvelrnda 5620	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → (𝐹‘𝑘) ∈ 𝐴)
5	4	adantlr 469	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ 𝑁) → (𝐹‘𝑘) ∈ 𝐴)
6		enumctlemm.n0	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑁)
7	3, 6	ffvelrnd 5621	. . . . 5 ⊢ (𝜑 → (𝐹‘∅) ∈ 𝐴)
8	7	ad2antrr 480	. . . 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 6468	. . . . 5 ⊢ ((𝑘 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑘 ∈ 𝑁)
13	9, 11, 12	syl2anc 409	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → DECID 𝑘 ∈ 𝑁)
14	5, 8, 13	ifcldadc 3549	. . 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 5639	. 2 ⊢ (𝜑 → 𝐺:ω⟶𝐴)
17		foelrn 5721	. . . . . 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 2227	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (𝑘 ∈ 𝑁 ↔ 𝑥 ∈ 𝑁))
20		fveq2 5486	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
21	19, 20	ifbieq1d 3542	. . . . . . . . . 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 4583	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑁 ∈ ω) → 𝑥 ∈ ω)
25	22, 23, 24	syl2anc 409	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ ω)
26	22	iftrued 3527	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)) = (𝐹‘𝑥))
27	3	ffvelrnda 5620	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐹‘𝑥) ∈ 𝐴)
28	26, 27	eqeltrd 2243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)) ∈ 𝐴)
29	15, 21, 25, 28	fvmptd3 5579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐺‘𝑥) = if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)))
30	29, 26	eqtrd 2198	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐺‘𝑥) = (𝐹‘𝑥))
31	30	eqeq2d 2177	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝑦 = (𝐺‘𝑥) ↔ 𝑦 = (𝐹‘𝑥)))
32	31	rexbidva 2463	. . . . . 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 4586	. . . . . . 7 ⊢ ω ∈ On
36	35	onelssi 4407	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ⊆ ω)
37		ssrexv 3207	. . . . . 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 2539	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥))
42		dffo3 5632	. 2 ⊢ (𝐺:ω–onto→𝐴 ↔ (𝐺:ω⟶𝐴 ∧ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥)))
43	16, 41, 42	sylanbrc 414	1 ⊢ (𝜑 → 𝐺:ω–onto→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 824   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445   ⊆ wss 3116  ∅c0 3409  ifcif 3520   ↦ cmpt 4043  ωcom 4567  ⟶wf 5184  –onto→wfo 5186  ‘cfv 5188

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196

This theorem is referenced by:  enumct  7080