Theorem enumctlemm 7180

Description: Lemma for enumct 7181. 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 5480	. . . . . . 7 ⊢ (𝐹:𝑁–onto→𝐴 → 𝐹:𝑁⟶𝐴)
3	1, 2	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑁⟶𝐴)
4	3	ffvelcdmda 5697	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → (𝐹‘𝑘) ∈ 𝐴)
5	4	adantlr 477	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑘 ∈ 𝑁) → (𝐹‘𝑘) ∈ 𝐴)
6		enumctlemm.n0	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑁)
7	3, 6	ffvelcdmd 5698	. . . . 5 ⊢ (𝜑 → (𝐹‘∅) ∈ 𝐴)
8	7	ad2antrr 488	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ ω) ∧ ¬ 𝑘 ∈ 𝑁) → (𝐹‘∅) ∈ 𝐴)
9		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → 𝑘 ∈ ω)
10		enumctlemm.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
11	10	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → 𝑁 ∈ ω)
12		nndcel 6558	. . . . 5 ⊢ ((𝑘 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑘 ∈ 𝑁)
13	9, 11, 12	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → DECID 𝑘 ∈ 𝑁)
14	5, 8, 13	ifcldadc 3590	. . 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 5716	. 2 ⊢ (𝜑 → 𝐺:ω⟶𝐴)
17		foelrn 5799	. . . . . 6 ⊢ ((𝐹:𝑁–onto→𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝑁 𝑦 = (𝐹‘𝑥))
18	1, 17	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝑁 𝑦 = (𝐹‘𝑥))
19		eleq1w 2257	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (𝑘 ∈ 𝑁 ↔ 𝑥 ∈ 𝑁))
20		fveq2 5558	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
21	19, 20	ifbieq1d 3583	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → if(𝑘 ∈ 𝑁, (𝐹‘𝑘), (𝐹‘∅)) = if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)))
22		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ 𝑁)
23	10	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑁 ∈ ω)
24		elnn 4642	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑁 ∈ ω) → 𝑥 ∈ ω)
25	22, 23, 24	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ ω)
26	22	iftrued 3568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)) = (𝐹‘𝑥))
27	3	ffvelcdmda 5697	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐹‘𝑥) ∈ 𝐴)
28	26, 27	eqeltrd 2273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)) ∈ 𝐴)
29	15, 21, 25, 28	fvmptd3 5655	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐺‘𝑥) = if(𝑥 ∈ 𝑁, (𝐹‘𝑥), (𝐹‘∅)))
30	29, 26	eqtrd 2229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐺‘𝑥) = (𝐹‘𝑥))
31	30	eqeq2d 2208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝑦 = (𝐺‘𝑥) ↔ 𝑦 = (𝐹‘𝑥)))
32	31	rexbidva 2494	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ 𝑁 𝑦 = (𝐺‘𝑥) ↔ ∃𝑥 ∈ 𝑁 𝑦 = (𝐹‘𝑥)))
33	32	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑥 ∈ 𝑁 𝑦 = (𝐺‘𝑥) ↔ ∃𝑥 ∈ 𝑁 𝑦 = (𝐹‘𝑥)))
34	18, 33	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝑁 𝑦 = (𝐺‘𝑥))
35		omelon 4645	. . . . . . 7 ⊢ ω ∈ On
36	35	onelssi 4464	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ⊆ ω)
37		ssrexv 3248	. . . . . 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 276	. . . 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 2570	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥))
42		dffo3 5709	. 2 ⊢ (𝐺:ω–onto→𝐴 ↔ (𝐺:ω⟶𝐴 ∧ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ω 𝑦 = (𝐺‘𝑥)))
43	16, 41, 42	sylanbrc 417	1 ⊢ (𝜑 → 𝐺:ω–onto→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476   ⊆ wss 3157  ∅c0 3450  ifcif 3561   ↦ cmpt 4094  ωcom 4626  ⟶wf 5254  –onto→wfo 5256  ‘cfv 5258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266

This theorem is referenced by:  enumct  7181