Theorem ctssdclemr 7229

Description: Lemma for ctssdc 7230. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)

Assertion

Ref	Expression
ctssdclemr	⊢ (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠))

Distinct variable groups:   𝐴,𝑓,𝑠   𝐴,𝑛,𝑠

Proof of Theorem ctssdclemr

Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		foeq1 5506	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
2	1	cbvexv 1943	. 2 ⊢ (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
3		id 19	. . . . . 6 ⊢ (𝑔:ω–onto→(𝐴 ⊔ 1o) → 𝑔:ω–onto→(𝐴 ⊔ 1o))
4		eqid 2206	. . . . . 6 ⊢ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}
5		eqid 2206	. . . . . 6 ⊢ (◡inl ∘ 𝑔) = (◡inl ∘ 𝑔)
6	3, 4, 5	ctssdccl 7228	. . . . 5 ⊢ (𝑔:ω–onto→(𝐴 ⊔ 1o) → ({𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ (◡inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
7		djulf1o 7175	. . . . . . . . 9 ⊢ inl:V–1-1-onto→({∅} × V)
8		f1ocnv 5547	. . . . . . . . 9 ⊢ (inl:V–1-1-onto→({∅} × V) → ◡inl:({∅} × V)–1-1-onto→V)
9		f1ofun 5536	. . . . . . . . 9 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
10	7, 8, 9	mp2b 8	. . . . . . . 8 ⊢ Fun ◡inl
11		vex 2776	. . . . . . . 8 ⊢ 𝑔 ∈ V
12		cofunexg 6207	. . . . . . . 8 ⊢ ((Fun ◡inl ∧ 𝑔 ∈ V) → (◡inl ∘ 𝑔) ∈ V)
13	10, 11, 12	mp2an 426	. . . . . . 7 ⊢ (◡inl ∘ 𝑔) ∈ V
14		foeq1 5506	. . . . . . 7 ⊢ (𝑓 = (◡inl ∘ 𝑔) → (𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 ↔ (◡inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴))
15	13, 14	spcev 2872	. . . . . 6 ⊢ ((◡inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 → ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴)
16	15	3anim2i 1189	. . . . 5 ⊢ (({𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ (◡inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}) → ({𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
17	6, 16	syl 14	. . . 4 ⊢ (𝑔:ω–onto→(𝐴 ⊔ 1o) → ({𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
18		omex 4649	. . . . . 6 ⊢ ω ∈ V
19	18	rabex 4196	. . . . 5 ⊢ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ∈ V
20		sseq1 3220	. . . . . 6 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑠 ⊆ ω ↔ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω))
21		foeq2 5507	. . . . . . 7 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑓:𝑠–onto→𝐴 ↔ 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴))
22	21	exbidv 1849	. . . . . 6 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (∃𝑓 𝑓:𝑠–onto→𝐴 ↔ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴))
23		eleq2 2270	. . . . . . . 8 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑛 ∈ 𝑠 ↔ 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
24	23	dcbid 840	. . . . . . 7 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (DECID 𝑛 ∈ 𝑠 ↔ DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
25	24	ralbidv 2507	. . . . . 6 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠 ↔ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
26	20, 22, 25	3anbi123d 1325	. . . . 5 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → ((𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠) ↔ ({𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)})))
27	19, 26	spcev 2872	. . . 4 ⊢ (({𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠))
28	17, 27	syl 14	. . 3 ⊢ (𝑔:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠))
29	28	exlimiv 1622	. 2 ⊢ (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠))
30	2, 29	sylbi 121	1 ⊢ (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠))

Syntax hints:   → wi 4  DECID wdc 836   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  {crab 2489  Vcvv 2773   ⊆ wss 3170  ∅c0 3464  {csn 3638  ωcom 4646   × cxp 4681  ^◡ccnv 4682   “ cima 4686   ∘ ccom 4687  Fun wfun 5274  –onto→wfo 5278  –1-1-onto→wf1o 5279  ‘cfv 5280  1_oc1o 6508   ⊔ cdju 7154  inlcinl 7162

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-1st 6239  df-2nd 6240  df-1o 6515  df-dju 7155  df-inl 7164  df-inr 7165

This theorem is referenced by:  ctssdc  7230