Theorem ctssdclemr 7105

Description: Lemma for ctssdc 7106. 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 5430	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
2	1	cbvexv 1918	. 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 2177	. . . . . 6 ⊢ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}
5		eqid 2177	. . . . . 6 ⊢ (◡inl ∘ 𝑔) = (◡inl ∘ 𝑔)
6	3, 4, 5	ctssdccl 7104	. . . . 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 7051	. . . . . . . . 9 ⊢ inl:V–1-1-onto→({∅} × V)
8		f1ocnv 5470	. . . . . . . . 9 ⊢ (inl:V–1-1-onto→({∅} × V) → ◡inl:({∅} × V)–1-1-onto→V)
9		f1ofun 5459	. . . . . . . . 9 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
10	7, 8, 9	mp2b 8	. . . . . . . 8 ⊢ Fun ◡inl
11		vex 2740	. . . . . . . 8 ⊢ 𝑔 ∈ V
12		cofunexg 6104	. . . . . . . 8 ⊢ ((Fun ◡inl ∧ 𝑔 ∈ V) → (◡inl ∘ 𝑔) ∈ V)
13	10, 11, 12	mp2an 426	. . . . . . 7 ⊢ (◡inl ∘ 𝑔) ∈ V
14		foeq1 5430	. . . . . . 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 2832	. . . . . 6 ⊢ ((◡inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴 → ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴)
16	15	3anim2i 1186	. . . . 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 4589	. . . . . 6 ⊢ ω ∈ V
19	18	rabex 4144	. . . . 5 ⊢ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ∈ V
20		sseq1 3178	. . . . . 6 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑠 ⊆ ω ↔ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} ⊆ ω))
21		foeq2 5431	. . . . . . 7 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑓:𝑠–onto→𝐴 ↔ 𝑓:{𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}–onto→𝐴))
22	21	exbidv 1825	. . . . . 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 2241	. . . . . . . 8 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (𝑛 ∈ 𝑠 ↔ 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
24	23	dcbid 838	. . . . . . 7 ⊢ (𝑠 = {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)} → (DECID 𝑛 ∈ 𝑠 ↔ DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔‘𝑡) ∈ (inl “ 𝐴)}))
25	24	ralbidv 2477	. . . . . 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 1312	. . . . 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 2832	. . . 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 1598	. 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 834   ∧ w3a 978   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  {crab 2459  Vcvv 2737   ⊆ wss 3129  ∅c0 3422  {csn 3591  ωcom 4586   × cxp 4621  ^◡ccnv 4622   “ cima 4626   ∘ ccom 4627  Fun wfun 5206  –onto→wfo 5210  –1-1-onto→wf1o 5211  ‘cfv 5212  1_oc1o 6404   ⊔ cdju 7030  inlcinl 7038

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-1o 6411  df-dju 7031  df-inl 7040  df-inr 7041

This theorem is referenced by:  ctssdc  7106