ssnnctlemct - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ssnnctlemct		GIF version

Theorem ssnnctlemct 12736

Description: Lemma for ssnnct 12737. 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 2267	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	dcbid 839	. . . 4 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑧 ∈ 𝐴))
3	2	cbvralv 2737	. . 3 ⊢ (∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ↔ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴)
4		imassrn 5030	. . . . 5 ⊢ (^◡𝐺 “ 𝐴) ⊆ ran ^◡𝐺
5		1z 9380	. . . . . . . . . 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 10532	. . . . . . . . . 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 9666	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
11		f1oeq3 5506	. . . . . . . . . 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 5529	. . . . . . . 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 5525	. . . . . . 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 3226	. . . 4 ⊢ (^◡𝐺 “ 𝐴) ⊆ ω
20		eleq1 2267	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘𝑦) → (𝑧 ∈ 𝐴 ↔ (𝐺‘𝑦) ∈ 𝐴))
21	20	dcbid 839	. . . . . . 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 5516	. . . . . . . . 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 5710	. . . . . . 7 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ ℕ)
27	21, 22, 26	rspcdva 2881	. . . . . 6 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → DECID (𝐺‘𝑦) ∈ 𝐴)
28		f1of1 5515	. . . . . . . . . 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 5832	. . . . . . . . 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 1354	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ω) → ((^◡𝐺‘(𝐺‘𝑦)) ∈ (^◡𝐺 “ 𝐴) ↔ (𝐺‘𝑦) ∈ 𝐴))
33		f1ocnvfv1 5836	. . . . . . . . . . 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 2273	. . . . . . . 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 839	. . . . . 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 2578	. . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑧 ∈ ℕ DECID 𝑧 ∈ 𝐴) → ∀𝑦 ∈ ω DECID 𝑦 ∈ (^◡𝐺 “ 𝐴))
41		ssomct 12735	. . . 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 9024	. . . . . 6 ⊢ ℕ ∈ V
45	44	ssex 4180	. . . . 5 ⊢ (𝐴 ⊆ ℕ → 𝐴 ∈ V)
46		f1ores 5531	. . . . . 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 6834	. . . . 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 12723	. . . 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 835 = wceq 1372 ∃wex 1514 ∈ wcel 2175 ∀wral 2483 Vcvv 2771 ⊆ wss 3165 class class class wbr 4043 ↦ cmpt 4104 ωcom 4636 ^◡ccnv 4672 ran crn 4674 ↾ cres 4675 “ cima 4676 ⟶wf 5264 –1-1→wf1 5265 –onto→wfo 5266 –1-1-onto→wf1o 5267 ‘cfv 5268 (class class class)co 5934 freccfrec 6466 1_oc1o 6485 ≈ cen 6815 ⊔ cdju 7121 1c1 7908 + caddc 7910 ℕcn 9018 ℤcz 9354 ℤ_≥cuz 9630

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-addcom 8007 ax-addass 8009 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-0id 8015 ax-rnegex 8016 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-ltadd 8023

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4338 df-iord 4411 df-on 4413 df-ilim 4414 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-1st 6216 df-2nd 6217 df-recs 6381 df-frec 6467 df-1o 6492 df-er 6610 df-en 6818 df-dju 7122 df-inl 7131 df-inr 7132 df-case 7168 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-inn 9019 df-n0 9278 df-z 9355 df-uz 9631

This theorem is referenced by: ssnnct 12737

W3C validator