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Theorem ssnnctlemct 13032

Description: Lemma for ssnnct 13033. The result. (Contributed by Jim Kingdon, 29-Sep-2024.)

**Hypothesis**
Ref	Expression
ssnnctlem.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 1)

**Assertion**
Ref	Expression
ssnnctlemct	⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1_o))

Distinct variable groups: 𝐴,𝑓 𝑥,𝐴 𝑓,𝐺 Allowed substitution hint: 𝐺(𝑥)

Proof of Theorem ssnnctlemct Dummy variables 𝑔 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2292	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	dcbid 843	. . . 4 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑧 ∈ 𝐴))
3	2	cbvralv 2765	. . 3 ⊢ (∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ↔ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴)
4		imassrn 5079	. . . . 5 ⊢ (^◡𝐺 “ 𝐴) ⊆ ran ^◡𝐺
5		1z 9483	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
6		id 19	. . . . . . . . . . 11 ⊢ (1 ∈ ℤ → 1 ∈ ℤ)
7		ssnnctlem.g	. . . . . . . . . . 11 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 1)
8	6, 7	frec2uzf1od 10640	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 𝐺:ω–1-1-onto→(ℤ_≥‘1))
9	5, 8	ax-mp 5	. . . . . . . . 9 ⊢ 𝐺:ω–1-1-onto→(ℤ_≥‘1)
10		nnuz 9770	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
11		f1oeq3 5564	. . . . . . . . . 10 ⊢ (ℕ = (ℤ_≥‘1) → (𝐺:ω–1-1-onto→ℕ ↔ 𝐺:ω–1-1-onto→(ℤ_≥‘1)))
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (𝐺:ω–1-1-onto→ℕ ↔ 𝐺:ω–1-1-onto→(ℤ_≥‘1))
13	9, 12	mpbir 146	. . . . . . . 8 ⊢ 𝐺:ω–1-1-onto→ℕ
14		f1ocnv 5587	. . . . . . . 8 ⊢ (𝐺:ω–1-1-onto→ℕ → ^◡𝐺:ℕ–1-1-onto→ω)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ ^◡𝐺:ℕ–1-1-onto→ω
16		dff1o5 5583	. . . . . . 7 ⊢ (^◡𝐺:ℕ–1-1-onto→ω ↔ (^◡𝐺:ℕ–1-1→ω ∧ ran ^◡𝐺 = ω))
17	15, 16	mpbi 145	. . . . . 6 ⊢ (^◡𝐺:ℕ–1-1→ω ∧ ran ^◡𝐺 = ω)
18	17	simpri 113	. . . . 5 ⊢ ran ^◡𝐺 = ω
19	4, 18	sseqtri 3258	. . . 4 ⊢ (^◡𝐺 “ 𝐴) ⊆ ω
20		eleq1 2292	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘𝑦) → (𝑧 ∈ 𝐴 ↔ (𝐺‘𝑦) ∈ 𝐴))
21	20	dcbid 843	. . . . . . 7 ⊢ (𝑧 = (𝐺‘𝑦) → (DECID 𝑧 ∈ 𝐴 ↔ DECID (𝐺‘𝑦) ∈ 𝐴))
22		simplr 528	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴)
23		f1of 5574	. . . . . . . . 9 ⊢ (𝐺:ω–1-1-onto→ℕ → 𝐺:ω⟶ℕ)
24	13, 23	mp1i 10	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → 𝐺:ω⟶ℕ)
25		simpr 110	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
26	24, 25	ffvelcdmd 5773	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ ℕ)
27	21, 22, 26	rspcdva 2912	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → DECID (𝐺‘𝑦) ∈ 𝐴)
28		f1of1 5573	. . . . . . . . . 10 ⊢ (^◡𝐺:ℕ–1-1-onto→ω → ^◡𝐺:ℕ–1-1→ω)
29	15, 28	ax-mp 5	. . . . . . . . 9 ⊢ ^◡𝐺:ℕ–1-1→ω
30		simpll 527	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → 𝐴 ⊆ ℕ)
31		f1elima 5903	. . . . . . . . 9 ⊢ ((^◡𝐺:ℕ–1-1→ω ∧ (𝐺‘𝑦) ∈ ℕ ∧ 𝐴 ⊆ ℕ) → ((^◡𝐺‘(𝐺‘𝑦)) ∈ (^◡𝐺 “ 𝐴) ↔ (𝐺‘𝑦) ∈ 𝐴))
32	29, 26, 30, 31	mp3an2i 1376	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → ((^◡𝐺‘(𝐺‘𝑦)) ∈ (^◡𝐺 “ 𝐴) ↔ (𝐺‘𝑦) ∈ 𝐴))
33		f1ocnvfv1 5907	. . . . . . . . . . 11 ⊢ ((𝐺:ω–1-1-onto→ℕ ∧ 𝑦 ∈ ω) → (^◡𝐺‘(𝐺‘𝑦)) = 𝑦)
34	13, 33	mpan 424	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (^◡𝐺‘(𝐺‘𝑦)) = 𝑦)
35	34	adantl 277	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → (^◡𝐺‘(𝐺‘𝑦)) = 𝑦)
36	35	eleq1d 2298	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → ((^◡𝐺‘(𝐺‘𝑦)) ∈ (^◡𝐺 “ 𝐴) ↔ 𝑦 ∈ (^◡𝐺 “ 𝐴)))
37	32, 36	bitr3d 190	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → ((𝐺‘𝑦) ∈ 𝐴 ↔ 𝑦 ∈ (^◡𝐺 “ 𝐴)))
38	37	dcbid 843	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → (DECID (𝐺‘𝑦) ∈ 𝐴 ↔ DECID 𝑦 ∈ (^◡𝐺 “ 𝐴)))
39	27, 38	mpbid 147	. . . . 5 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → DECID 𝑦 ∈ (^◡𝐺 “ 𝐴))
40	39	ralrimiva 2603	. . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) → ∀𝑦 ∈ ω DECID 𝑦 ∈ (^◡𝐺 “ 𝐴))
41		ssomct 13031	. . . 4 ⊢ (((^◡𝐺 “ 𝐴) ⊆ ω ∧ ∀𝑦 ∈ ω DECID 𝑦 ∈ (^◡𝐺 “ 𝐴)) → ∃𝑔 𝑔:ω–onto→((^◡𝐺 “ 𝐴) ⊔ 1_o))
42	19, 40, 41	sylancr 414	. . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) → ∃𝑔 𝑔:ω–onto→((^◡𝐺 “ 𝐴) ⊔ 1_o))
43	3, 42	sylan2b 287	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → ∃𝑔 𝑔:ω–onto→((^◡𝐺 “ 𝐴) ⊔ 1_o))
44		nnex 9127	. . . . . 6 ⊢ ℕ ∈ V
45	44	ssex 4221	. . . . 5 ⊢ (𝐴 ⊆ ℕ → 𝐴 ∈ V)
46		f1ores 5589	. . . . . 6 ⊢ ((^◡𝐺:ℕ–1-1→ω ∧ 𝐴 ⊆ ℕ) → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
47	29, 46	mpan 424	. . . . 5 ⊢ (𝐴 ⊆ ℕ → (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴))
48		f1oeng 6916	. . . . 5 ⊢ ((𝐴 ∈ V ∧ (^◡𝐺 ↾ 𝐴):𝐴–1-1-onto→(^◡𝐺 “ 𝐴)) → 𝐴 ≈ (^◡𝐺 “ 𝐴))
49	45, 47, 48	syl2anc 411	. . . 4 ⊢ (𝐴 ⊆ ℕ → 𝐴 ≈ (^◡𝐺 “ 𝐴))
50		enct 13019	. . . 4 ⊢ (𝐴 ≈ (^◡𝐺 “ 𝐴) → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1_o) ↔ ∃𝑔 𝑔:ω–onto→((^◡𝐺 “ 𝐴) ⊔ 1_o)))
51	49, 50	syl 14	. . 3 ⊢ (𝐴 ⊆ ℕ → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1_o) ↔ ∃𝑔 𝑔:ω–onto→((^◡𝐺 “ 𝐴) ⊔ 1_o)))
52	51	adantr 276	. 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1_o) ↔ ∃𝑔 𝑔:ω–onto→((^◡𝐺 “ 𝐴) ⊔ 1_o)))
53	43, 52	mpbird 167	1 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1_o))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 839 = wceq 1395 ∃wex 1538 ∈ wcel 2200 ∀wral 2508 Vcvv 2799 ⊆ wss 3197 class class class wbr 4083 ↦ cmpt 4145 ωcom 4682 ^◡ccnv 4718 ran crn 4720 ↾ cres 4721 “ cima 4722 ⟶wf 5314 –1-1→wf1 5315 –onto→wfo 5316 –1-1-onto→wf1o 5317 ‘cfv 5318 (class class class)co 6007 freccfrec 6542 1_oc1o 6561 ≈ cen 6893 ⊔ cdju 7215 1c1 8011 + caddc 8013 ℕcn 9121 ℤcz 9457 ℤ_≥cuz 9733

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-addcom 8110 ax-addass 8112 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-0id 8118 ax-rnegex 8119 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-ltadd 8126

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-1o 6568 df-er 6688 df-en 6896 df-dju 7216 df-inl 7225 df-inr 7226 df-case 7262 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-inn 9122 df-n0 9381 df-z 9458 df-uz 9734

This theorem is referenced by: ssnnct 13033

W3C validator