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Theorem ctssdccl 7104
Description: A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7106 but expressed in terms of classes rather than . (Contributed by Jim Kingdon, 30-Oct-2023.)
Hypotheses
Ref Expression
ctssdccl.f (𝜑𝐹:ω–onto→(𝐴 ⊔ 1o))
ctssdccl.s 𝑆 = {𝑥 ∈ ω ∣ (𝐹𝑥) ∈ (inl “ 𝐴)}
ctssdccl.g 𝐺 = (inl ∘ 𝐹)
Assertion
Ref Expression
ctssdccl (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑆))
Distinct variable groups:   𝑥,𝐴   𝑛,𝐹,𝑥   𝑛,𝐺   𝑆,𝑛   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑛)   𝑆(𝑥)   𝐺(𝑥)

Proof of Theorem ctssdccl
Dummy variables 𝑚 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ctssdccl.s . . . 4 𝑆 = {𝑥 ∈ ω ∣ (𝐹𝑥) ∈ (inl “ 𝐴)}
2 ssrab2 3240 . . . 4 {𝑥 ∈ ω ∣ (𝐹𝑥) ∈ (inl “ 𝐴)} ⊆ ω
31, 2eqsstri 3187 . . 3 𝑆 ⊆ ω
43a1i 9 . 2 (𝜑𝑆 ⊆ ω)
5 djulf1o 7051 . . . . . . 7 inl:V–1-1-onto→({∅} × V)
6 f1ocnv 5470 . . . . . . 7 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
7 f1ofun 5459 . . . . . . 7 (inl:({∅} × V)–1-1-onto→V → Fun inl)
85, 6, 7mp2b 8 . . . . . 6 Fun inl
9 ctssdccl.f . . . . . . 7 (𝜑𝐹:ω–onto→(𝐴 ⊔ 1o))
10 fofun 5435 . . . . . . 7 (𝐹:ω–onto→(𝐴 ⊔ 1o) → Fun 𝐹)
119, 10syl 14 . . . . . 6 (𝜑 → Fun 𝐹)
12 funco 5252 . . . . . . 7 ((Fun inl ∧ Fun 𝐹) → Fun (inl ∘ 𝐹))
13 ctssdccl.g . . . . . . . 8 𝐺 = (inl ∘ 𝐹)
1413funeqi 5233 . . . . . . 7 (Fun 𝐺 ↔ Fun (inl ∘ 𝐹))
1512, 14sylibr 134 . . . . . 6 ((Fun inl ∧ Fun 𝐹) → Fun 𝐺)
168, 11, 15sylancr 414 . . . . 5 (𝜑 → Fun 𝐺)
17 fof 5434 . . . . . . . . . . . 12 (𝐹:ω–onto→(𝐴 ⊔ 1o) → 𝐹:ω⟶(𝐴 ⊔ 1o))
189, 17syl 14 . . . . . . . . . . 11 (𝜑𝐹:ω⟶(𝐴 ⊔ 1o))
1918fdmd 5368 . . . . . . . . . 10 (𝜑 → dom 𝐹 = ω)
2019eleq2d 2247 . . . . . . . . 9 (𝜑 → (𝑛 ∈ dom 𝐹𝑛 ∈ ω))
2120anbi1d 465 . . . . . . . 8 (𝜑 → ((𝑛 ∈ dom 𝐹 ∧ (𝐹𝑛) ∈ dom inl) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ dom inl)))
22 dmcoss 4892 . . . . . . . . . . . 12 dom (inl ∘ 𝐹) ⊆ dom 𝐹
2322sseli 3151 . . . . . . . . . . 11 (𝑛 ∈ dom (inl ∘ 𝐹) → 𝑛 ∈ dom 𝐹)
2423pm4.71ri 392 . . . . . . . . . 10 (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹𝑛 ∈ dom (inl ∘ 𝐹)))
25 dmfco 5580 . . . . . . . . . . 11 ((Fun 𝐹𝑛 ∈ dom 𝐹) → (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝐹𝑛) ∈ dom inl))
2625pm5.32da 452 . . . . . . . . . 10 (Fun 𝐹 → ((𝑛 ∈ dom 𝐹𝑛 ∈ dom (inl ∘ 𝐹)) ↔ (𝑛 ∈ dom 𝐹 ∧ (𝐹𝑛) ∈ dom inl)))
2724, 26bitrid 192 . . . . . . . . 9 (Fun 𝐹 → (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹 ∧ (𝐹𝑛) ∈ dom inl)))
2811, 27syl 14 . . . . . . . 8 (𝜑 → (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹 ∧ (𝐹𝑛) ∈ dom inl)))
29 simpr 110 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → (𝐹𝑛) ∈ (inl “ 𝐴))
30 imassrn 4977 . . . . . . . . . . . . . 14 (inl “ 𝐴) ⊆ ran inl
3130sseli 3151 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (inl “ 𝐴) → (𝐹𝑛) ∈ ran inl)
3231adantl 277 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → (𝐹𝑛) ∈ ran inl)
33 df-rn 4634 . . . . . . . . . . . . 13 ran inl = dom inl
3433eleq2i 2244 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ ran inl ↔ (𝐹𝑛) ∈ dom inl)
3532, 34sylib 122 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → (𝐹𝑛) ∈ dom inl)
3629, 352thd 175 . . . . . . . . . 10 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ dom inl))
37 djuin 7057 . . . . . . . . . . . . . 14 ((inl “ 𝐴) ∩ (inr “ 1o)) = ∅
38 disjel 3477 . . . . . . . . . . . . . 14 ((((inl “ 𝐴) ∩ (inr “ 1o)) = ∅ ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
3937, 38mpan 424 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (inl “ 𝐴) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
4039con2i 627 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ (inr “ 1o) → ¬ (𝐹𝑛) ∈ (inl “ 𝐴))
4140adantl 277 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ¬ (𝐹𝑛) ∈ (inl “ 𝐴))
42 djuin 7057 . . . . . . . . . . . . . . . 16 ((inl “ V) ∩ (inr “ 1o)) = ∅
43 disjel 3477 . . . . . . . . . . . . . . . 16 ((((inl “ V) ∩ (inr “ 1o)) = ∅ ∧ (𝐹𝑛) ∈ (inl “ V)) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
4442, 43mpan 424 . . . . . . . . . . . . . . 15 ((𝐹𝑛) ∈ (inl “ V) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
45 dfrn4 5085 . . . . . . . . . . . . . . 15 ran inl = (inl “ V)
4644, 45eleq2s 2272 . . . . . . . . . . . . . 14 ((𝐹𝑛) ∈ ran inl → ¬ (𝐹𝑛) ∈ (inr “ 1o))
4746con2i 627 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (inr “ 1o) → ¬ (𝐹𝑛) ∈ ran inl)
4847adantl 277 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ¬ (𝐹𝑛) ∈ ran inl)
4948, 34sylnib 676 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ¬ (𝐹𝑛) ∈ dom inl)
5041, 492falsed 702 . . . . . . . . . 10 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ dom inl))
5118ffvelcdmda 5647 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ω) → (𝐹𝑛) ∈ (𝐴 ⊔ 1o))
52 djuun 7060 . . . . . . . . . . . . 13 ((inl “ 𝐴) ∪ (inr “ 1o)) = (𝐴 ⊔ 1o)
5352eleq2i 2244 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)) ↔ (𝐹𝑛) ∈ (𝐴 ⊔ 1o))
5451, 53sylibr 134 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ω) → (𝐹𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)))
55 elun 3276 . . . . . . . . . . 11 ((𝐹𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)) ↔ ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ (𝐹𝑛) ∈ (inr “ 1o)))
5654, 55sylib 122 . . . . . . . . . 10 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ (𝐹𝑛) ∈ (inr “ 1o)))
5736, 50, 56mpjaodan 798 . . . . . . . . 9 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ dom inl))
5857pm5.32da 452 . . . . . . . 8 (𝜑 → ((𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ dom inl)))
5921, 28, 583bitr4d 220 . . . . . . 7 (𝜑 → (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴))))
6013dmeqi 4824 . . . . . . . 8 dom 𝐺 = dom (inl ∘ 𝐹)
6160eleq2i 2244 . . . . . . 7 (𝑛 ∈ dom 𝐺𝑛 ∈ dom (inl ∘ 𝐹))
62 fveq2 5511 . . . . . . . . 9 (𝑥 = 𝑛 → (𝐹𝑥) = (𝐹𝑛))
6362eleq1d 2246 . . . . . . . 8 (𝑥 = 𝑛 → ((𝐹𝑥) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ (inl “ 𝐴)))
6463, 1elrab2 2896 . . . . . . 7 (𝑛𝑆 ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴)))
6559, 61, 643bitr4g 223 . . . . . 6 (𝜑 → (𝑛 ∈ dom 𝐺𝑛𝑆))
6665eqrdv 2175 . . . . 5 (𝜑 → dom 𝐺 = 𝑆)
67 df-fn 5215 . . . . 5 (𝐺 Fn 𝑆 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝑆))
6816, 66, 67sylanbrc 417 . . . 4 (𝜑𝐺 Fn 𝑆)
6913fveq1i 5512 . . . . . . 7 (𝐺𝑚) = ((inl ∘ 𝐹)‘𝑚)
7018adantr 276 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1o))
71 fveq2 5511 . . . . . . . . . . . . 13 (𝑥 = 𝑚 → (𝐹𝑥) = (𝐹𝑚))
7271eleq1d 2246 . . . . . . . . . . . 12 (𝑥 = 𝑚 → ((𝐹𝑥) ∈ (inl “ 𝐴) ↔ (𝐹𝑚) ∈ (inl “ 𝐴)))
7372, 1elrab2 2896 . . . . . . . . . . 11 (𝑚𝑆 ↔ (𝑚 ∈ ω ∧ (𝐹𝑚) ∈ (inl “ 𝐴)))
7473biimpi 120 . . . . . . . . . 10 (𝑚𝑆 → (𝑚 ∈ ω ∧ (𝐹𝑚) ∈ (inl “ 𝐴)))
7574adantl 277 . . . . . . . . 9 ((𝜑𝑚𝑆) → (𝑚 ∈ ω ∧ (𝐹𝑚) ∈ (inl “ 𝐴)))
7675simpld 112 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝑚 ∈ ω)
77 fvco3 5583 . . . . . . . 8 ((𝐹:ω⟶(𝐴 ⊔ 1o) ∧ 𝑚 ∈ ω) → ((inl ∘ 𝐹)‘𝑚) = (inl‘(𝐹𝑚)))
7870, 76, 77syl2anc 411 . . . . . . 7 ((𝜑𝑚𝑆) → ((inl ∘ 𝐹)‘𝑚) = (inl‘(𝐹𝑚)))
7969, 78eqtrid 2222 . . . . . 6 ((𝜑𝑚𝑆) → (𝐺𝑚) = (inl‘(𝐹𝑚)))
80 f1ofun 5459 . . . . . . . . . 10 (inl:V–1-1-onto→({∅} × V) → Fun inl)
815, 80ax-mp 5 . . . . . . . . 9 Fun inl
82 fvelima 5563 . . . . . . . . 9 ((Fun inl ∧ (𝐹𝑚) ∈ (inl “ 𝐴)) → ∃𝑧𝐴 (inl‘𝑧) = (𝐹𝑚))
8381, 82mpan 424 . . . . . . . 8 ((𝐹𝑚) ∈ (inl “ 𝐴) → ∃𝑧𝐴 (inl‘𝑧) = (𝐹𝑚))
8475, 83simpl2im 386 . . . . . . 7 ((𝜑𝑚𝑆) → ∃𝑧𝐴 (inl‘𝑧) = (𝐹𝑚))
85 simprr 531 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘𝑧) = (𝐹𝑚))
8685fveq2d 5515 . . . . . . . 8 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘(inl‘𝑧)) = (inl‘(𝐹𝑚)))
87 vex 2740 . . . . . . . . . 10 𝑧 ∈ V
88 f1ocnvfv1 5772 . . . . . . . . . 10 ((inl:V–1-1-onto→({∅} × V) ∧ 𝑧 ∈ V) → (inl‘(inl‘𝑧)) = 𝑧)
895, 87, 88mp2an 426 . . . . . . . . 9 (inl‘(inl‘𝑧)) = 𝑧
90 simprl 529 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → 𝑧𝐴)
9189, 90eqeltrid 2264 . . . . . . . 8 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘(inl‘𝑧)) ∈ 𝐴)
9286, 91eqeltrrd 2255 . . . . . . 7 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘(𝐹𝑚)) ∈ 𝐴)
9384, 92rexlimddv 2599 . . . . . 6 ((𝜑𝑚𝑆) → (inl‘(𝐹𝑚)) ∈ 𝐴)
9479, 93eqeltrd 2254 . . . . 5 ((𝜑𝑚𝑆) → (𝐺𝑚) ∈ 𝐴)
9594ralrimiva 2550 . . . 4 (𝜑 → ∀𝑚𝑆 (𝐺𝑚) ∈ 𝐴)
96 ffnfv 5670 . . . 4 (𝐺:𝑆𝐴 ↔ (𝐺 Fn 𝑆 ∧ ∀𝑚𝑆 (𝐺𝑚) ∈ 𝐴))
9768, 95, 96sylanbrc 417 . . 3 (𝜑𝐺:𝑆𝐴)
98 djulcl 7044 . . . . . . . 8 (𝑚𝐴 → (inl‘𝑚) ∈ (𝐴 ⊔ 1o))
99 foelrn 5748 . . . . . . . . . 10 ((𝐹:ω–onto→(𝐴 ⊔ 1o) ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹𝑦))
1009, 99sylan 283 . . . . . . . . 9 ((𝜑 ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹𝑦))
101 df-rex 2461 . . . . . . . . 9 (∃𝑦 ∈ ω (inl‘𝑚) = (𝐹𝑦) ↔ ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)))
102100, 101sylib 122 . . . . . . . 8 ((𝜑 ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)))
10398, 102sylan2 286 . . . . . . 7 ((𝜑𝑚𝐴) → ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)))
104 fveq2 5511 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
105104eleq1d 2246 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ (inl “ 𝐴) ↔ (𝐹𝑦) ∈ (inl “ 𝐴)))
106 simprl 529 . . . . . . . . . . . 12 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → 𝑦 ∈ ω)
107 simprr 531 . . . . . . . . . . . . 13 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (inl‘𝑚) = (𝐹𝑦))
108 vex 2740 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
109 f1odm 5461 . . . . . . . . . . . . . . . . 17 (inl:V–1-1-onto→({∅} × V) → dom inl = V)
1105, 109ax-mp 5 . . . . . . . . . . . . . . . 16 dom inl = V
111108, 110eleqtrri 2253 . . . . . . . . . . . . . . 15 𝑚 ∈ dom inl
112 funfvima 5743 . . . . . . . . . . . . . . 15 ((Fun inl ∧ 𝑚 ∈ dom inl) → (𝑚𝐴 → (inl‘𝑚) ∈ (inl “ 𝐴)))
11381, 111, 112mp2an 426 . . . . . . . . . . . . . 14 (𝑚𝐴 → (inl‘𝑚) ∈ (inl “ 𝐴))
114113ad2antlr 489 . . . . . . . . . . . . 13 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (inl‘𝑚) ∈ (inl “ 𝐴))
115107, 114eqeltrrd 2255 . . . . . . . . . . . 12 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (𝐹𝑦) ∈ (inl “ 𝐴))
116105, 106, 115elrabd 2895 . . . . . . . . . . 11 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → 𝑦 ∈ {𝑥 ∈ ω ∣ (𝐹𝑥) ∈ (inl “ 𝐴)})
117116, 1eleqtrrdi 2271 . . . . . . . . . 10 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → 𝑦𝑆)
118117, 107jca 306 . . . . . . . . 9 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (𝑦𝑆 ∧ (inl‘𝑚) = (𝐹𝑦)))
119118ex 115 . . . . . . . 8 ((𝜑𝑚𝐴) → ((𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)) → (𝑦𝑆 ∧ (inl‘𝑚) = (𝐹𝑦))))
120119eximdv 1880 . . . . . . 7 ((𝜑𝑚𝐴) → (∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)) → ∃𝑦(𝑦𝑆 ∧ (inl‘𝑚) = (𝐹𝑦))))
121103, 120mpd 13 . . . . . 6 ((𝜑𝑚𝐴) → ∃𝑦(𝑦𝑆 ∧ (inl‘𝑚) = (𝐹𝑦)))
122 df-rex 2461 . . . . . 6 (∃𝑦𝑆 (inl‘𝑚) = (𝐹𝑦) ↔ ∃𝑦(𝑦𝑆 ∧ (inl‘𝑚) = (𝐹𝑦)))
123121, 122sylibr 134 . . . . 5 ((𝜑𝑚𝐴) → ∃𝑦𝑆 (inl‘𝑚) = (𝐹𝑦))
124 f1ocnvfv1 5772 . . . . . . . . . 10 ((inl:V–1-1-onto→({∅} × V) ∧ 𝑚 ∈ V) → (inl‘(inl‘𝑚)) = 𝑚)
1255, 108, 124mp2an 426 . . . . . . . . 9 (inl‘(inl‘𝑚)) = 𝑚
126 simpr 110 . . . . . . . . . 10 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → (inl‘𝑚) = (𝐹𝑦))
127126fveq2d 5515 . . . . . . . . 9 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → (inl‘(inl‘𝑚)) = (inl‘(𝐹𝑦)))
128125, 127eqtr3id 2224 . . . . . . . 8 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → 𝑚 = (inl‘(𝐹𝑦)))
12913fveq1i 5512 . . . . . . . . . 10 (𝐺𝑦) = ((inl ∘ 𝐹)‘𝑦)
13018ad2antrr 488 . . . . . . . . . . 11 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1o))
1313sseli 3151 . . . . . . . . . . . 12 (𝑦𝑆𝑦 ∈ ω)
132131adantl 277 . . . . . . . . . . 11 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → 𝑦 ∈ ω)
133 fvco3 5583 . . . . . . . . . . 11 ((𝐹:ω⟶(𝐴 ⊔ 1o) ∧ 𝑦 ∈ ω) → ((inl ∘ 𝐹)‘𝑦) = (inl‘(𝐹𝑦)))
134130, 132, 133syl2anc 411 . . . . . . . . . 10 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → ((inl ∘ 𝐹)‘𝑦) = (inl‘(𝐹𝑦)))
135129, 134eqtrid 2222 . . . . . . . . 9 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → (𝐺𝑦) = (inl‘(𝐹𝑦)))
136135adantr 276 . . . . . . . 8 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → (𝐺𝑦) = (inl‘(𝐹𝑦)))
137128, 136eqtr4d 2213 . . . . . . 7 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → 𝑚 = (𝐺𝑦))
138137ex 115 . . . . . 6 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → ((inl‘𝑚) = (𝐹𝑦) → 𝑚 = (𝐺𝑦)))
139138reximdva 2579 . . . . 5 ((𝜑𝑚𝐴) → (∃𝑦𝑆 (inl‘𝑚) = (𝐹𝑦) → ∃𝑦𝑆 𝑚 = (𝐺𝑦)))
140123, 139mpd 13 . . . 4 ((𝜑𝑚𝐴) → ∃𝑦𝑆 𝑚 = (𝐺𝑦))
141140ralrimiva 2550 . . 3 (𝜑 → ∀𝑚𝐴𝑦𝑆 𝑚 = (𝐺𝑦))
142 dffo3 5659 . . 3 (𝐺:𝑆onto𝐴 ↔ (𝐺:𝑆𝐴 ∧ ∀𝑚𝐴𝑦𝑆 𝑚 = (𝐺𝑦)))
14397, 141, 142sylanbrc 417 . 2 (𝜑𝐺:𝑆onto𝐴)
14453, 55bitr3i 186 . . . . . . 7 ((𝐹𝑛) ∈ (𝐴 ⊔ 1o) ↔ ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ (𝐹𝑛) ∈ (inr “ 1o)))
14551, 144sylib 122 . . . . . 6 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ (𝐹𝑛) ∈ (inr “ 1o)))
14640orim2i 761 . . . . . 6 (((𝐹𝑛) ∈ (inl “ 𝐴) ∨ (𝐹𝑛) ∈ (inr “ 1o)) → ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ ¬ (𝐹𝑛) ∈ (inl “ 𝐴)))
147145, 146syl 14 . . . . 5 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ ¬ (𝐹𝑛) ∈ (inl “ 𝐴)))
148 df-dc 835 . . . . 5 (DECID (𝐹𝑛) ∈ (inl “ 𝐴) ↔ ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ ¬ (𝐹𝑛) ∈ (inl “ 𝐴)))
149147, 148sylibr 134 . . . 4 ((𝜑𝑛 ∈ ω) → DECID (𝐹𝑛) ∈ (inl “ 𝐴))
150 ibar 301 . . . . . . 7 (𝑛 ∈ ω → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴))))
151150adantl 277 . . . . . 6 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴))))
152151, 64bitr4di 198 . . . . 5 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ 𝑛𝑆))
153152dcbid 838 . . . 4 ((𝜑𝑛 ∈ ω) → (DECID (𝐹𝑛) ∈ (inl “ 𝐴) ↔ DECID 𝑛𝑆))
154149, 153mpbid 147 . . 3 ((𝜑𝑛 ∈ ω) → DECID 𝑛𝑆)
155154ralrimiva 2550 . 2 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
1564, 143, 1553jca 1177 1 (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑆))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  {crab 2459  Vcvv 2737  cun 3127  cin 3128  wss 3129  c0 3422  {csn 3591  ωcom 4586   × cxp 4621  ccnv 4622  dom cdm 4623  ran crn 4624  cima 4626  ccom 4627  Fun wfun 5206   Fn wfn 5207  wf 5208  ontowfo 5210  1-1-ontowf1o 5211  cfv 5212  1oc1o 6404  cdju 7030  inlcinl 7038  inrcinr 7039
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-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430
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-rab 2464  df-v 2739  df-sbc 2963  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-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  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:  ctssdclemr  7105  ctiunct  12424
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