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Theorem ssnnctlemct 13066

Description: Lemma for ssnnct 13067. 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 2294	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	dcbid 845	. . . 4 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑧 ∈ 𝐴))
3	2	cbvralv 2767	. . 3 ⊢ (∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ↔ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴)
4		imassrn 5087	. . . . 5 ⊢ (^◡𝐺 “ 𝐴) ⊆ ran ^◡𝐺
5		1z 9504	. . . . . . . . . 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 10667	. . . . . . . . . 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 9791	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
11		f1oeq3 5573	. . . . . . . . . 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 5596	. . . . . . . 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 5592	. . . . . . 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 3261	. . . 4 ⊢ (^◡𝐺 “ 𝐴) ⊆ ω
20		eleq1 2294	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘𝑦) → (𝑧 ∈ 𝐴 ↔ (𝐺‘𝑦) ∈ 𝐴))
21	20	dcbid 845	. . . . . . 7 ⊢ (𝑧 = (𝐺‘𝑦) → (DECID 𝑧 ∈ 𝐴 ↔ DECID (𝐺‘𝑦) ∈ 𝐴))
22		simplr 529	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴)
23		f1of 5583	. . . . . . . . 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 5783	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ ℕ)
27	21, 22, 26	rspcdva 2915	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → DECID (𝐺‘𝑦) ∈ 𝐴)
28		f1of1 5582	. . . . . . . . . 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 5913	. . . . . . . . 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 1378	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → ((^◡𝐺‘(𝐺‘𝑦)) ∈ (^◡𝐺 “ 𝐴) ↔ (𝐺‘𝑦) ∈ 𝐴))
33		f1ocnvfv1 5917	. . . . . . . . . . 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 2300	. . . . . . . 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 845	. . . . . 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 2605	. . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) → ∀𝑦 ∈ ω DECID 𝑦 ∈ (^◡𝐺 “ 𝐴))
41		ssomct 13065	. . . 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 9148	. . . . . 6 ⊢ ℕ ∈ V
45	44	ssex 4226	. . . . 5 ⊢ (𝐴 ⊆ ℕ → 𝐴 ∈ V)
46		f1ores 5598	. . . . . 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 6929	. . . . 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 13053	. . . 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 841 = wceq 1397 ∃wex 1540 ∈ wcel 2202 ∀wral 2510 Vcvv 2802 ⊆ wss 3200 class class class wbr 4088 ↦ cmpt 4150 ωcom 4688 ^◡ccnv 4724 ran crn 4726 ↾ cres 4727 “ cima 4728 ⟶wf 5322 –1-1→wf1 5323 –onto→wfo 5324 –1-1-onto→wf1o 5325 ‘cfv 5326 (class class class)co 6017 freccfrec 6555 1_oc1o 6574 ≈ cen 6906 ⊔ cdju 7235 1c1 8032 + caddc 8034 ℕcn 9142 ℤcz 9478 ℤ_≥cuz 9754

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-addcom 8131 ax-addass 8133 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-0id 8139 ax-rnegex 8140 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-ltadd 8147

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-1o 6581 df-er 6701 df-en 6909 df-dju 7236 df-inl 7245 df-inr 7246 df-case 7282 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-inn 9143 df-n0 9402 df-z 9479 df-uz 9755

This theorem is referenced by: ssnnct 13067

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