Theorem ctm 7125

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 5513	. . . . . . . . . . 11 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
2		f1of 5475	. . . . . . . . . . 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 5428	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (1o × {𝑥}):1o⟶𝐴)
5	4	adantr 276	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (1o × {𝑥}):1o⟶𝐴)
6	3, 5	casef 7104	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴)
7		ffun 5382	. . . . . . . . 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 2754	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10	9	a1i 9	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓 ∈ V)
11		cofunexg 6127	. . . . . . . 8 ⊢ ((Fun case(( I ↾ 𝐴), (1o × {𝑥})) ∧ 𝑓 ∈ V) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V)
12	8, 10, 11	syl2anc 411	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V)
13		fof 5452	. . . . . . . . . 10 ⊢ (𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝑓:ω⟶(𝐴 ⊔ 1o))
14	13	adantl 277	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓:ω⟶(𝐴 ⊔ 1o))
15		fco 5395	. . . . . . . . 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 411	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω⟶𝐴)
17		simplr 528	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → 𝑓:ω–onto→(𝐴 ⊔ 1o))
18		djulcl 7067	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
19	18	adantl 277	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
20		foelrn 5768	. . . . . . . . . . 11 ⊢ ((𝑓:ω–onto→(𝐴 ⊔ 1o) ∧ (inl‘𝑦) ∈ (𝐴 ⊔ 1o)) → ∃𝑧 ∈ ω (inl‘𝑦) = (𝑓‘𝑧))
21	17, 19, 20	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ ω (inl‘𝑦) = (𝑓‘𝑧))
22		fofn 5454	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝑓 Fn ω)
23	22	ad4antlr 495	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑓 Fn ω)
24		simplr 528	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑧 ∈ ω)
25		fvco2 5600	. . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . 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 110	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → (inl‘𝑦) = (𝑓‘𝑧))
28	27	fveq2d 5533	. . . . . . . . . . . . . 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 5347	. . . . . . . . . . . . . . . 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 2754	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
32	31	fconst6 5429	. . . . . . . . . . . . . . . 16 ⊢ (1o × {𝑥}):1o⟶V
33		ffun 5382	. . . . . . . . . . . . . . . 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 534	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓‘𝑧)) → 𝑦 ∈ 𝐴)
36	30, 34, 35	caseinl 7107	. . . . . . . . . . . . . 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 2227	. . . . . . . . . . . . 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 5724	. . . . . . . . . . . . . 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 2222	. . . . . . . . . . . 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 115	. . . . . . . . . . 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 2591	. . . . . . . . . 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 2562	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
45		dffo3 5678	. . . . . . . 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 417	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto→𝐴)
47		foeq1 5448	. . . . . . . 8 ⊢ (𝑔 = (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) → (𝑔:ω–onto→𝐴 ↔ (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto→𝐴))
48	47	spcegv 2839	. . . . . . 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 115	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
51	50	exlimiv 1608	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
52	51	exlimdv 1829	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→𝐴))
53		foeq1 5448	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→𝐴 ↔ 𝑔:ω–onto→𝐴))
54	53	cbvexv 1929	. . 3 ⊢ (∃𝑓 𝑓:ω–onto→𝐴 ↔ ∃𝑔 𝑔:ω–onto→𝐴)
55	52, 54	imbitrrdi 162	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑓 𝑓:ω–onto→𝐴))
56		ctmlemr 7124	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))
57	55, 56	impbid 129	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto→𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1363  ∃wex 1502   ∈ wcel 2159  ∀wral 2467  ∃wrex 2468  Vcvv 2751  {csn 3606   I cid 4302  ωcom 4603   × cxp 4638   ↾ cres 4642   ∘ ccom 4644  Fun wfun 5224   Fn wfn 5225  ⟶wf 5226  –onto→wfo 5228  –1-1-onto→wf1o 5229  ‘cfv 5230  1_oc1o 6427   ⊔ cdju 7053  inlcinl 7061  casecdjucase 7099

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-iinf 4601

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-1st 6158  df-2nd 6159  df-1o 6434  df-dju 7054  df-inl 7063  df-inr 7064  df-case 7100

This theorem is referenced by:  ctssdc  7129  enumct  7131  omct  7133  unbendc  12472  pw1nct  15136