Theorem ctssdclemr 7295

Description: Lemma for ctssdc 7296. 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 5549	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
2	1	cbvexv 1965	. 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 2229	. . . . . 6 ⊢ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}
5		eqid 2229	. . . . . 6 ⊢ (◡inl ∘ 𝑔) = (◡inl ∘ 𝑔)
6	3, 4, 5	ctssdccl 7294	. . . . 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 7241	. . . . . . . . 9 ⊢ inl:V–1-1-onto→({∅} × V)
8		f1ocnv 5590	. . . . . . . . 9 ⊢ (inl:V–1-1-onto→({∅} × V) → ◡inl:({∅} × V)–1-1-onto→V)
9		f1ofun 5579	. . . . . . . . 9 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
10	7, 8, 9	mp2b 8	. . . . . . . 8 ⊢ Fun ◡inl
11		vex 2802	. . . . . . . 8 ⊢ 𝑔 ∈ V
12		cofunexg 6263	. . . . . . . 8 ⊢ ((Fun ◡inl ∧ 𝑔 ∈ V) → (◡inl ∘ 𝑔) ∈ V)
13	10, 11, 12	mp2an 426	. . . . . . 7 ⊢ (◡inl ∘ 𝑔) ∈ V
14		foeq1 5549	. . . . . . 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 2898	. . . . . 6 ⊢ ((◡inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 → ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴)
16	15	3anim2i 1210	. . . . 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 4686	. . . . . 6 ⊢ ω ∈ V
19	18	rabex 4229	. . . . 5 ⊢ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ∈ V
20		sseq1 3247	. . . . . 6 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑠 ⊆ ω ↔ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω))
21		foeq2 5550	. . . . . . 7 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑓:𝑠–onto→𝐴 ↔ 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴))
22	21	exbidv 1871	. . . . . 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 2293	. . . . . . . 8 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑛 ∈ 𝑠 ↔ 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
24	23	dcbid 843	. . . . . . 7 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (DECID 𝑛 ∈ 𝑠 ↔ DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
25	24	ralbidv 2530	. . . . . 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 1346	. . . . 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 2898	. . . 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 1644	. 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 839   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  {crab 2512  Vcvv 2799   ⊆ wss 3197  ∅c0 3491  {csn 3666  ωcom 4683   × cxp 4718  ^◡ccnv 4719   “ cima 4723   ∘ ccom 4724  Fun wfun 5315  –onto→wfo 5319  –1-1-onto→wf1o 5320  ‘cfv 5321  1_oc1o 6566   ⊔ cdju 7220  inlcinl 7228

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-iinf 4681

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-1st 6295  df-2nd 6296  df-1o 6573  df-dju 7221  df-inl 7230  df-inr 7231

This theorem is referenced by:  ctssdc  7296