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Theorem ssnnctlemct 13286

Description: Lemma for ssnnct 13287. 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 2297	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	dcbid 846	. . . 4 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑧 ∈ 𝐴))
3	2	cbvralv 2780	. . 3 ⊢ (∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ↔ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴)
4		imassrn 5118	. . . . 5 ⊢ (^◡𝐺 “ 𝐴) ⊆ ran ^◡𝐺
5		1z 9624	. . . . . . . . . 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 10796	. . . . . . . . . 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 9912	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
11		f1oeq3 5610	. . . . . . . . . 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 5633	. . . . . . . 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 5629	. . . . . . 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 3276	. . . 4 ⊢ (^◡𝐺 “ 𝐴) ⊆ ω
20		eleq1 2297	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘𝑦) → (𝑧 ∈ 𝐴 ↔ (𝐺‘𝑦) ∈ 𝐴))
21	20	dcbid 846	. . . . . . 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 5620	. . . . . . . . 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 5819	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ ℕ)
27	21, 22, 26	rspcdva 2928	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → DECID (𝐺‘𝑦) ∈ 𝐴)
28		f1of1 5619	. . . . . . . . . 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 5953	. . . . . . . . 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 1379	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → ((^◡𝐺‘(𝐺‘𝑦)) ∈ (^◡𝐺 “ 𝐴) ↔ (𝐺‘𝑦) ∈ 𝐴))
33		f1ocnvfv1 5957	. . . . . . . . . . 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 2303	. . . . . . . 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 846	. . . . . 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 2617	. . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) → ∀𝑦 ∈ ω DECID 𝑦 ∈ (^◡𝐺 “ 𝐴))
41		ssomct 13285	. . . 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 9264	. . . . . 6 ⊢ ℕ ∈ V
45	44	ssex 4253	. . . . 5 ⊢ (𝐴 ⊆ ℕ → 𝐴 ∈ V)
46		f1ores 5635	. . . . . 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 7010	. . . . 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 13273	. . . 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 842 = wceq 1398 ∃wex 1541 ∈ wcel 2205 ∀wral 2522 Vcvv 2815 ⊆ wss 3214 class class class wbr 4115 ↦ cmpt 4177 ωcom 4718 ^◡ccnv 4754 ran crn 4756 ↾ cres 4757 “ cima 4758 ⟶wf 5354 –1-1→wf1 5355 –onto→wfo 5356 –1-1-onto→wf1o 5357 ‘cfv 5358 (class class class)co 6059 freccfrec 6635 1_oc1o 6654 ≈ cen 6987 ⊔ cdju 7342 1c1 8145 + caddc 8147 ℕcn 9258 ℤcz 9598 ℤ_≥cuz 9875

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4231 ax-sep 4234 ax-nul 4242 ax-pow 4293 ax-pr 4328 ax-un 4560 ax-setind 4665 ax-iinf 4716 ax-cnex 8235 ax-resscn 8236 ax-1cn 8237 ax-1re 8238 ax-icn 8239 ax-addcl 8240 ax-addrcl 8241 ax-mulcl 8242 ax-addcom 8244 ax-addass 8246 ax-distr 8248 ax-i2m1 8249 ax-0lt1 8250 ax-0id 8252 ax-rnegex 8253 ax-cnre 8255 ax-pre-ltirr 8256 ax-pre-ltwlin 8257 ax-pre-lttrn 8258 ax-pre-ltadd 8260

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3626 df-pw 3677 df-sn 3701 df-pr 3702 df-op 3704 df-uni 3921 df-int 3956 df-iun 3999 df-br 4116 df-opab 4178 df-mpt 4179 df-tr 4215 df-id 4420 df-iord 4493 df-on 4495 df-ilim 4496 df-suc 4498 df-iom 4719 df-xp 4761 df-rel 4762 df-cnv 4763 df-co 4764 df-dm 4765 df-rn 4766 df-res 4767 df-ima 4768 df-iota 5318 df-fun 5360 df-fn 5361 df-f 5362 df-f1 5363 df-fo 5364 df-f1o 5365 df-fv 5366 df-riota 6012 df-ov 6062 df-oprab 6063 df-mpo 6064 df-1st 6348 df-2nd 6349 df-recs 6550 df-frec 6636 df-1o 6661 df-er 6781 df-en 6990 df-dju 7343 df-inl 7352 df-inr 7353 df-case 7389 df-pnf 8327 df-mnf 8328 df-xr 8329 df-ltxr 8330 df-le 8331 df-sub 8464 df-neg 8465 df-inn 9259 df-n0 9518 df-z 9599 df-uz 9876

This theorem is referenced by: ssnnct 13287

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