Theorem ctm 7002

Description: Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)

Assertion

Ref	Expression
ctm	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto→𝐴))

Distinct variable group:   𝐴,𝑓,𝑥

Proof of Theorem ctm

Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 5413	. . . . . . . . . . 11 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
2		f1of 5375	. . . . . . . . . . 11 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 → ( I ↾ 𝐴):𝐴⟶𝐴)
3	1, 2	mp1i 10	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ( I ↾ 𝐴):𝐴⟶𝐴)
4		fconst6g 5329	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (1o × {𝑥}):1o⟶𝐴)
5	4	adantr 274	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (1o × {𝑥}):1o⟶𝐴)
6	3, 5	casef 6981	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴)
7		ffun 5283	. . . . . . . . 9 ⊢ (case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴 → Fun case(( I ↾ 𝐴), (1o × {𝑥})))
8	6, 7	syl 14	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → Fun case(( I ↾ 𝐴), (1o × {𝑥})))
9		vex 2692	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10	9	a1i 9	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓 ∈ V)
11		cofunexg 6017	. . . . . . . 8 ⊢ ((Fun case(( I ↾ 𝐴), (1o × {𝑥})) ∧ 𝑓 ∈ V) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V)
12	8, 10, 11	syl2anc 409	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V)
13		fof 5353	. . . . . . . . . 10 ⊢ (𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝑓:ω⟶(𝐴 ⊔ 1o))
14	13	adantl 275	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓:ω⟶(𝐴 ⊔ 1o))
15		fco 5296	. . . . . . . . 9 ⊢ ((case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴 ∧ 𝑓:ω⟶(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω⟶𝐴)
16	6, 14, 15	syl2anc 409	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω⟶𝐴)
17		simplr 520	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → 𝑓:ω–onto→(𝐴 ⊔ 1o))
18		djulcl 6944	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
19	18	adantl 275	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
20		foelrn 5662	. . . . . . . . . . 11 ⊢ ((𝑓:ω–onto→(𝐴 ⊔ 1o) ∧ (inl‘𝑦) ∈ (𝐴 ⊔ 1o)) → ∃𝑧 ∈ ω (inl‘𝑦) = (𝑓‘𝑧))
21	17, 19, 20	syl2anc 409	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ ω (inl‘𝑦) = (𝑓‘𝑧))
22		fofn 5355	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝑓 Fn ω)
23	22	ad4antlr 487	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑓 Fn ω)
24		simplr 520	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑧 ∈ ω)
25		fvco2 5498	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn ω ∧ 𝑧 ∈ ω) → ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧) = (case(( I ↾ 𝐴), (1o × {𝑥}))‘(𝑓‘𝑧)))
26	23, 24, 25	syl2anc 409	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧) = (case(( I ↾ 𝐴), (1o × {𝑥}))‘(𝑓‘𝑧)))
27		simpr 109	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → (inl‘𝑦) = (𝑓‘𝑧))
28	27	fveq2d 5433	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → (case(( I ↾ 𝐴), (1o × {𝑥}))‘(inl‘𝑦)) = (case(( I ↾ 𝐴), (1o × {𝑥}))‘(𝑓‘𝑧)))
29		fnresi 5248	. . . . . . . . . . . . . . . 16 ⊢ ( I ↾ 𝐴) Fn 𝐴
30	29	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → ( I ↾ 𝐴) Fn 𝐴)
31		vex 2692	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
32	31	fconst6 5330	. . . . . . . . . . . . . . . 16 ⊢ (1o × {𝑥}):1o⟶V
33		ffun 5283	. . . . . . . . . . . . . . . 16 ⊢ ((1o × {𝑥}):1o⟶V → Fun (1o × {𝑥}))
34	32, 33	mp1i 10	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → Fun (1o × {𝑥}))
35		simpllr 524	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑦 ∈ 𝐴)
36	30, 34, 35	caseinl 6984	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → (case(( I ↾ 𝐴), (1o × {𝑥}))‘(inl‘𝑦)) = (( I ↾ 𝐴)‘𝑦))
37	26, 28, 36	3eqtr2d 2179	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧) = (( I ↾ 𝐴)‘𝑦))
38		fvresi 5621	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
39	35, 38	syl 14	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
40	37, 39	eqtr2d 2174	. . . . . . . . . . . 12 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
41	40	ex 114	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) → ((inl‘𝑦) = (𝑓‘𝑧) → 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧)))
42	41	reximdva 2537	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ ω (inl‘𝑦) = (𝑓‘𝑧) → ∃𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧)))
43	21, 42	mpd 13	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
44	43	ralrimiva 2508	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
45		dffo3 5575	. . . . . . . 8 ⊢ ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto→𝐴 ↔ ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω⟶𝐴 ∧ ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧)))
46	16, 44, 45	sylanbrc 414	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto→𝐴)
47		foeq1 5349	. . . . . . . 8 ⊢ (𝑔 = (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) → (𝑔:ω–onto→𝐴 ↔ (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto→𝐴))
48	47	spcegv 2777	. . . . . . 7 ⊢ ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V → ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto→𝐴 → ∃𝑔 𝑔:ω–onto→𝐴))
49	12, 46, 48	sylc 62	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∃𝑔 𝑔:ω–onto→𝐴)
50	49	ex 114	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
51	50	exlimiv 1578	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
52	51	exlimdv 1792	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
53		foeq1 5349	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→𝐴 ↔ 𝑔:ω–onto→𝐴))
54	53	cbvexv 1891	. . 3 ⊢ (∃𝑓 𝑓:ω–onto→𝐴 ↔ ∃𝑔 𝑔:ω–onto→𝐴)
55	52, 54	syl6ibr 161	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑓 𝑓:ω–onto→𝐴))
56		ctmlemr 7001	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))
57	55, 56	impbid 128	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto→𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2689  {csn 3532   I cid 4218  ωcom 4512   × cxp 4545   ↾ cres 4549   ∘ ccom 4551  Fun wfun 5125   Fn wfn 5126  ⟶wf 5127  –onto→wfo 5129  –1-1-onto→wf1o 5130  ‘cfv 5131  1_oc1o 6314   ⊔ cdju 6930  inlcinl 6938  casecdjucase 6976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-dju 6931  df-inl 6940  df-inr 6941  df-case 6977

This theorem is referenced by:  ctssdc  7006  enumct  7008  omct  7010  pw1nct  13371