Theorem ctm 6871

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 5326	. . . . . . . . . . 11 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
2		f1of 5288	. . . . . . . . . . 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 5244	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (1o × {𝑥}):1o⟶𝐴)
5	4	adantr 271	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (1o × {𝑥}):1o⟶𝐴)
6	3, 5	casef 6859	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴)
7		ffun 5198	. . . . . . . . 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 2636	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10	9	a1i 9	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓 ∈ V)
11		cofunexg 5920	. . . . . . . 8 ⊢ ((Fun case(( I ↾ 𝐴), (1o × {𝑥})) ∧ 𝑓 ∈ V) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V)
12	8, 10, 11	syl2anc 404	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V)
13		fof 5268	. . . . . . . . . 10 ⊢ (𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝑓:ω⟶(𝐴 ⊔ 1o))
14	13	adantl 272	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓:ω⟶(𝐴 ⊔ 1o))
15		fco 5211	. . . . . . . . 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 404	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω⟶𝐴)
17		simplr 498	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → 𝑓:ω–onto→(𝐴 ⊔ 1o))
18		djulcl 6823	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
19	18	adantl 272	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
20		foelrn 5570	. . . . . . . . . . 11 ⊢ ((𝑓:ω–onto→(𝐴 ⊔ 1o) ∧ (inl‘𝑦) ∈ (𝐴 ⊔ 1o)) → ∃𝑧 ∈ ω (inl‘𝑦) = (𝑓‘𝑧))
21	17, 19, 20	syl2anc 404	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ ω (inl‘𝑦) = (𝑓‘𝑧))
22		fofn 5270	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝑓 Fn ω)
23	22	ad4antlr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑓 Fn ω)
24		simplr 498	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑧 ∈ ω)
25		fvco2 5408	. . . . . . . . . . . . . . 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 404	. . . . . . . . . . . . . 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 5344	. . . . . . . . . . . . . 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 5165	. . . . . . . . . . . . . . . 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 2636	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
32	31	fconst6 5245	. . . . . . . . . . . . . . . 16 ⊢ (1o × {𝑥}):1o⟶V
33		ffun 5198	. . . . . . . . . . . . . . . 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 502	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑦 ∈ 𝐴)
36	30, 34, 35	caseinl 6862	. . . . . . . . . . . . . 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 2133	. . . . . . . . . . . . 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 5529	. . . . . . . . . . . . . 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 2128	. . . . . . . . . . . 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 2487	. . . . . . . . . 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 2458	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
45		dffo3 5485	. . . . . . . 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 409	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto→𝐴)
47		foeq1 5264	. . . . . . . 8 ⊢ (𝑔 = (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) → (𝑔:ω–onto→𝐴 ↔ (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto→𝐴))
48	47	spcegv 2721	. . . . . . 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 1541	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
52	51	exlimdv 1754	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
53		foeq1 5264	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→𝐴 ↔ 𝑔:ω–onto→𝐴))
54	53	cbvexv 1850	. . 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 6870	. 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 1296  ∃wex 1433   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371  Vcvv 2633  {csn 3466   I cid 4139  ωcom 4433   × cxp 4465   ↾ cres 4469   ∘ ccom 4471  Fun wfun 5043   Fn wfn 5044  ⟶wf 5045  –onto→wfo 5047  –1-1-onto→wf1o 5048  ‘cfv 5049  1_oc1o 6212   ⊔ cdju 6810  inlcinl 6817  casecdjucase 6854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950  df-1o 6219  df-dju 6811  df-inl 6819  df-inr 6820  df-case 6855

This theorem is referenced by:  enumct  6873