Theorem ctssdclemr 7402

Description: Lemma for ctssdc 7403. 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 5585	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
2	1	cbvexv 1968	. 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 2232	. . . . . 6 ⊢ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}
5		eqid 2232	. . . . . 6 ⊢ (◡inl ∘ 𝑔) = (◡inl ∘ 𝑔)
6	3, 4, 5	ctssdccl 7401	. . . . 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 7348	. . . . . . . . 9 ⊢ inl:V–1-1-onto→({∅} × V)
8		f1ocnv 5626	. . . . . . . . 9 ⊢ (inl:V–1-1-onto→({∅} × V) → ◡inl:({∅} × V)–1-1-onto→V)
9		f1ofun 5615	. . . . . . . . 9 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
10	7, 8, 9	mp2b 8	. . . . . . . 8 ⊢ Fun ◡inl
11		vex 2815	. . . . . . . 8 ⊢ 𝑔 ∈ V
12		cofunexg 6301	. . . . . . . 8 ⊢ ((Fun ◡inl ∧ 𝑔 ∈ V) → (◡inl ∘ 𝑔) ∈ V)
13	10, 11, 12	mp2an 426	. . . . . . 7 ⊢ (◡inl ∘ 𝑔) ∈ V
14		foeq1 5585	. . . . . . 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 2911	. . . . . 6 ⊢ ((◡inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 → ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴)
16	15	3anim2i 1213	. . . . 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 4714	. . . . . 6 ⊢ ω ∈ V
19	18	rabex 4255	. . . . 5 ⊢ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ∈ V
20		sseq1 3260	. . . . . 6 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑠 ⊆ ω ↔ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω))
21		foeq2 5586	. . . . . . 7 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑓:𝑠–onto→𝐴 ↔ 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴))
22	21	exbidv 1874	. . . . . 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 2296	. . . . . . . 8 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑛 ∈ 𝑠 ↔ 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
24	23	dcbid 846	. . . . . . 7 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (DECID 𝑛 ∈ 𝑠 ↔ DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
25	24	ralbidv 2542	. . . . . 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 1349	. . . . 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 2911	. . . 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 1647	. 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 842   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  {crab 2524  Vcvv 2812   ⊆ wss 3210  ∅c0 3507  {csn 3688  ωcom 4711   × cxp 4746  ^◡ccnv 4747   “ cima 4751   ∘ ccom 4752  Fun wfun 5345  –onto→wfo 5349  –1-1-onto→wf1o 5350  ‘cfv 5351  1_oc1o 6639   ⊔ cdju 7327  inlcinl 7335

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-1st 6333  df-2nd 6334  df-1o 6646  df-dju 7328  df-inl 7337  df-inr 7338

This theorem is referenced by:  ctssdc  7403