Theorem ctm 7070

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 5469	. . . . . . . . . . 11 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
2		f1of 5431	. . . . . . . . . . 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 5385	. . . . . . . . . . 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 7049	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴)
7		ffun 5339	. . . . . . . . 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 2728	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10	9	a1i 9	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓 ∈ V)
11		cofunexg 6076	. . . . . . . 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 5409	. . . . . . . . . 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 5352	. . . . . . . . 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 7012	. . . . . . . . . . . 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 5720	. . . . . . . . . . 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 5411	. . . . . . . . . . . . . . . 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 5554	. . . . . . . . . . . . . . 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 5489	. . . . . . . . . . . . . 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 5304	. . . . . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
32	31	fconst6 5386	. . . . . . . . . . . . . . . 16 ⊢ (1o × {𝑥}):1o⟶V
33		ffun 5339	. . . . . . . . . . . . . . . 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 7052	. . . . . . . . . . . . . 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 2204	. . . . . . . . . . . . 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 5677	. . . . . . . . . . . . . 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 2199	. . . . . . . . . . . 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 2567	. . . . . . . . . 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 2538	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
45		dffo3 5631	. . . . . . . 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 5405	. . . . . . . 8 ⊢ (𝑔 = (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) → (𝑔:ω–onto→𝐴 ↔ (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto→𝐴))
48	47	spcegv 2813	. . . . . . 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 1586	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
52	51	exlimdv 1807	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
53		foeq1 5405	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→𝐴 ↔ 𝑔:ω–onto→𝐴))
54	53	cbvexv 1906	. . 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 7069	. 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 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2443  ∃wrex 2444  Vcvv 2725  {csn 3575   I cid 4265  ωcom 4566   × cxp 4601   ↾ cres 4605   ∘ ccom 4607  Fun wfun 5181   Fn wfn 5182  ⟶wf 5183  –onto→wfo 5185  –1-1-onto→wf1o 5186  ‘cfv 5187  1_oc1o 6373   ⊔ cdju 6998  inlcinl 7006  casecdjucase 7044

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-iinf 4564

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-ral 2448  df-rex 2449  df-reu 2450  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-iord 4343  df-on 4345  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-1st 6105  df-2nd 6106  df-1o 6380  df-dju 6999  df-inl 7008  df-inr 7009  df-case 7045

This theorem is referenced by:  ctssdc  7074  enumct  7076  omct  7078  unbendc  12383  pw1nct  13843