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Theorem ctssdccl 6962
Description: A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 6964 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 3150 . . . 4 {𝑥 ∈ ω ∣ (𝐹𝑥) ∈ (inl “ 𝐴)} ⊆ ω
31, 2eqsstri 3097 . . 3 𝑆 ⊆ ω
43a1i 9 . 2 (𝜑𝑆 ⊆ ω)
5 djulf1o 6909 . . . . . . 7 inl:V–1-1-onto→({∅} × V)
6 f1ocnv 5346 . . . . . . 7 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
7 f1ofun 5335 . . . . . . 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 5314 . . . . . . 7 (𝐹:ω–onto→(𝐴 ⊔ 1o) → Fun 𝐹)
119, 10syl 14 . . . . . 6 (𝜑 → Fun 𝐹)
12 funco 5131 . . . . . . 7 ((Fun inl ∧ Fun 𝐹) → Fun (inl ∘ 𝐹))
13 ctssdccl.g . . . . . . . 8 𝐺 = (inl ∘ 𝐹)
1413funeqi 5112 . . . . . . 7 (Fun 𝐺 ↔ Fun (inl ∘ 𝐹))
1512, 14sylibr 133 . . . . . 6 ((Fun inl ∧ Fun 𝐹) → Fun 𝐺)
168, 11, 15sylancr 408 . . . . 5 (𝜑 → Fun 𝐺)
17 fof 5313 . . . . . . . . . . . 12 (𝐹:ω–onto→(𝐴 ⊔ 1o) → 𝐹:ω⟶(𝐴 ⊔ 1o))
189, 17syl 14 . . . . . . . . . . 11 (𝜑𝐹:ω⟶(𝐴 ⊔ 1o))
1918fdmd 5247 . . . . . . . . . 10 (𝜑 → dom 𝐹 = ω)
2019eleq2d 2185 . . . . . . . . 9 (𝜑 → (𝑛 ∈ dom 𝐹𝑛 ∈ ω))
2120anbi1d 458 . . . . . . . 8 (𝜑 → ((𝑛 ∈ dom 𝐹 ∧ (𝐹𝑛) ∈ dom inl) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ dom inl)))
22 dmcoss 4776 . . . . . . . . . . . 12 dom (inl ∘ 𝐹) ⊆ dom 𝐹
2322sseli 3061 . . . . . . . . . . 11 (𝑛 ∈ dom (inl ∘ 𝐹) → 𝑛 ∈ dom 𝐹)
2423pm4.71ri 387 . . . . . . . . . 10 (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹𝑛 ∈ dom (inl ∘ 𝐹)))
25 dmfco 5455 . . . . . . . . . . 11 ((Fun 𝐹𝑛 ∈ dom 𝐹) → (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝐹𝑛) ∈ dom inl))
2625pm5.32da 445 . . . . . . . . . 10 (Fun 𝐹 → ((𝑛 ∈ dom 𝐹𝑛 ∈ dom (inl ∘ 𝐹)) ↔ (𝑛 ∈ dom 𝐹 ∧ (𝐹𝑛) ∈ dom inl)))
2724, 26syl5bb 191 . . . . . . . . 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 109 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → (𝐹𝑛) ∈ (inl “ 𝐴))
30 imassrn 4860 . . . . . . . . . . . . . 14 (inl “ 𝐴) ⊆ ran inl
3130sseli 3061 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (inl “ 𝐴) → (𝐹𝑛) ∈ ran inl)
3231adantl 273 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → (𝐹𝑛) ∈ ran inl)
33 df-rn 4518 . . . . . . . . . . . . 13 ran inl = dom inl
3433eleq2i 2182 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ ran inl ↔ (𝐹𝑛) ∈ dom inl)
3532, 34sylib 121 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → (𝐹𝑛) ∈ dom inl)
3629, 352thd 174 . . . . . . . . . 10 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ dom inl))
37 djuin 6915 . . . . . . . . . . . . . 14 ((inl “ 𝐴) ∩ (inr “ 1o)) = ∅
38 disjel 3385 . . . . . . . . . . . . . 14 ((((inl “ 𝐴) ∩ (inr “ 1o)) = ∅ ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
3937, 38mpan 418 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (inl “ 𝐴) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
4039con2i 599 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ (inr “ 1o) → ¬ (𝐹𝑛) ∈ (inl “ 𝐴))
4140adantl 273 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ¬ (𝐹𝑛) ∈ (inl “ 𝐴))
42 djuin 6915 . . . . . . . . . . . . . . . 16 ((inl “ V) ∩ (inr “ 1o)) = ∅
43 disjel 3385 . . . . . . . . . . . . . . . 16 ((((inl “ V) ∩ (inr “ 1o)) = ∅ ∧ (𝐹𝑛) ∈ (inl “ V)) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
4442, 43mpan 418 . . . . . . . . . . . . . . 15 ((𝐹𝑛) ∈ (inl “ V) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
45 dfrn4 4967 . . . . . . . . . . . . . . 15 ran inl = (inl “ V)
4644, 45eleq2s 2210 . . . . . . . . . . . . . 14 ((𝐹𝑛) ∈ ran inl → ¬ (𝐹𝑛) ∈ (inr “ 1o))
4746con2i 599 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (inr “ 1o) → ¬ (𝐹𝑛) ∈ ran inl)
4847adantl 273 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ¬ (𝐹𝑛) ∈ ran inl)
4948, 34sylnib 648 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ¬ (𝐹𝑛) ∈ dom inl)
5041, 492falsed 674 . . . . . . . . . 10 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ dom inl))
5118ffvelrnda 5521 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ω) → (𝐹𝑛) ∈ (𝐴 ⊔ 1o))
52 djuun 6918 . . . . . . . . . . . . 13 ((inl “ 𝐴) ∪ (inr “ 1o)) = (𝐴 ⊔ 1o)
5352eleq2i 2182 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)) ↔ (𝐹𝑛) ∈ (𝐴 ⊔ 1o))
5451, 53sylibr 133 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ω) → (𝐹𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)))
55 elun 3185 . . . . . . . . . . 11 ((𝐹𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)) ↔ ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ (𝐹𝑛) ∈ (inr “ 1o)))
5654, 55sylib 121 . . . . . . . . . 10 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ (𝐹𝑛) ∈ (inr “ 1o)))
5736, 50, 56mpjaodan 770 . . . . . . . . 9 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ dom inl))
5857pm5.32da 445 . . . . . . . 8 (𝜑 → ((𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ dom inl)))
5921, 28, 583bitr4d 219 . . . . . . 7 (𝜑 → (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴))))
6013dmeqi 4708 . . . . . . . 8 dom 𝐺 = dom (inl ∘ 𝐹)
6160eleq2i 2182 . . . . . . 7 (𝑛 ∈ dom 𝐺𝑛 ∈ dom (inl ∘ 𝐹))
62 fveq2 5387 . . . . . . . . 9 (𝑥 = 𝑛 → (𝐹𝑥) = (𝐹𝑛))
6362eleq1d 2184 . . . . . . . 8 (𝑥 = 𝑛 → ((𝐹𝑥) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ (inl “ 𝐴)))
6463, 1elrab2 2814 . . . . . . 7 (𝑛𝑆 ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴)))
6559, 61, 643bitr4g 222 . . . . . 6 (𝜑 → (𝑛 ∈ dom 𝐺𝑛𝑆))
6665eqrdv 2113 . . . . 5 (𝜑 → dom 𝐺 = 𝑆)
67 df-fn 5094 . . . . 5 (𝐺 Fn 𝑆 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝑆))
6816, 66, 67sylanbrc 411 . . . 4 (𝜑𝐺 Fn 𝑆)
6913fveq1i 5388 . . . . . . 7 (𝐺𝑚) = ((inl ∘ 𝐹)‘𝑚)
7018adantr 272 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1o))
71 fveq2 5387 . . . . . . . . . . . . 13 (𝑥 = 𝑚 → (𝐹𝑥) = (𝐹𝑚))
7271eleq1d 2184 . . . . . . . . . . . 12 (𝑥 = 𝑚 → ((𝐹𝑥) ∈ (inl “ 𝐴) ↔ (𝐹𝑚) ∈ (inl “ 𝐴)))
7372, 1elrab2 2814 . . . . . . . . . . 11 (𝑚𝑆 ↔ (𝑚 ∈ ω ∧ (𝐹𝑚) ∈ (inl “ 𝐴)))
7473biimpi 119 . . . . . . . . . 10 (𝑚𝑆 → (𝑚 ∈ ω ∧ (𝐹𝑚) ∈ (inl “ 𝐴)))
7574adantl 273 . . . . . . . . 9 ((𝜑𝑚𝑆) → (𝑚 ∈ ω ∧ (𝐹𝑚) ∈ (inl “ 𝐴)))
7675simpld 111 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝑚 ∈ ω)
77 fvco3 5458 . . . . . . . 8 ((𝐹:ω⟶(𝐴 ⊔ 1o) ∧ 𝑚 ∈ ω) → ((inl ∘ 𝐹)‘𝑚) = (inl‘(𝐹𝑚)))
7870, 76, 77syl2anc 406 . . . . . . 7 ((𝜑𝑚𝑆) → ((inl ∘ 𝐹)‘𝑚) = (inl‘(𝐹𝑚)))
7969, 78syl5eq 2160 . . . . . 6 ((𝜑𝑚𝑆) → (𝐺𝑚) = (inl‘(𝐹𝑚)))
80 f1ofun 5335 . . . . . . . . . 10 (inl:V–1-1-onto→({∅} × V) → Fun inl)
815, 80ax-mp 5 . . . . . . . . 9 Fun inl
82 fvelima 5439 . . . . . . . . 9 ((Fun inl ∧ (𝐹𝑚) ∈ (inl “ 𝐴)) → ∃𝑧𝐴 (inl‘𝑧) = (𝐹𝑚))
8381, 82mpan 418 . . . . . . . 8 ((𝐹𝑚) ∈ (inl “ 𝐴) → ∃𝑧𝐴 (inl‘𝑧) = (𝐹𝑚))
8475, 83simpl2im 381 . . . . . . 7 ((𝜑𝑚𝑆) → ∃𝑧𝐴 (inl‘𝑧) = (𝐹𝑚))
85 simprr 504 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘𝑧) = (𝐹𝑚))
8685fveq2d 5391 . . . . . . . 8 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘(inl‘𝑧)) = (inl‘(𝐹𝑚)))
87 vex 2661 . . . . . . . . . 10 𝑧 ∈ V
88 f1ocnvfv1 5644 . . . . . . . . . 10 ((inl:V–1-1-onto→({∅} × V) ∧ 𝑧 ∈ V) → (inl‘(inl‘𝑧)) = 𝑧)
895, 87, 88mp2an 420 . . . . . . . . 9 (inl‘(inl‘𝑧)) = 𝑧
90 simprl 503 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → 𝑧𝐴)
9189, 90eqeltrid 2202 . . . . . . . 8 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘(inl‘𝑧)) ∈ 𝐴)
9286, 91eqeltrrd 2193 . . . . . . 7 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘(𝐹𝑚)) ∈ 𝐴)
9384, 92rexlimddv 2529 . . . . . 6 ((𝜑𝑚𝑆) → (inl‘(𝐹𝑚)) ∈ 𝐴)
9479, 93eqeltrd 2192 . . . . 5 ((𝜑𝑚𝑆) → (𝐺𝑚) ∈ 𝐴)
9594ralrimiva 2480 . . . 4 (𝜑 → ∀𝑚𝑆 (𝐺𝑚) ∈ 𝐴)
96 ffnfv 5544 . . . 4 (𝐺:𝑆𝐴 ↔ (𝐺 Fn 𝑆 ∧ ∀𝑚𝑆 (𝐺𝑚) ∈ 𝐴))
9768, 95, 96sylanbrc 411 . . 3 (𝜑𝐺:𝑆𝐴)
98 djulcl 6902 . . . . . . . 8 (𝑚𝐴 → (inl‘𝑚) ∈ (𝐴 ⊔ 1o))
99 foelrn 5620 . . . . . . . . . 10 ((𝐹:ω–onto→(𝐴 ⊔ 1o) ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹𝑦))
1009, 99sylan 279 . . . . . . . . 9 ((𝜑 ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹𝑦))
101 df-rex 2397 . . . . . . . . 9 (∃𝑦 ∈ ω (inl‘𝑚) = (𝐹𝑦) ↔ ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)))
102100, 101sylib 121 . . . . . . . 8 ((𝜑 ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)))
10398, 102sylan2 282 . . . . . . 7 ((𝜑𝑚𝐴) → ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)))
104 fveq2 5387 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
105104eleq1d 2184 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ (inl “ 𝐴) ↔ (𝐹𝑦) ∈ (inl “ 𝐴)))
106 simprl 503 . . . . . . . . . . . 12 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → 𝑦 ∈ ω)
107 simprr 504 . . . . . . . . . . . . 13 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (inl‘𝑚) = (𝐹𝑦))
108 vex 2661 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
109 f1odm 5337 . . . . . . . . . . . . . . . . 17 (inl:V–1-1-onto→({∅} × V) → dom inl = V)
1105, 109ax-mp 5 . . . . . . . . . . . . . . . 16 dom inl = V
111108, 110eleqtrri 2191 . . . . . . . . . . . . . . 15 𝑚 ∈ dom inl
112 funfvima 5615 . . . . . . . . . . . . . . 15 ((Fun inl ∧ 𝑚 ∈ dom inl) → (𝑚𝐴 → (inl‘𝑚) ∈ (inl “ 𝐴)))
11381, 111, 112mp2an 420 . . . . . . . . . . . . . 14 (𝑚𝐴 → (inl‘𝑚) ∈ (inl “ 𝐴))
114113ad2antlr 478 . . . . . . . . . . . . 13 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (inl‘𝑚) ∈ (inl “ 𝐴))
115107, 114eqeltrrd 2193 . . . . . . . . . . . 12 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (𝐹𝑦) ∈ (inl “ 𝐴))
116105, 106, 115elrabd 2813 . . . . . . . . . . 11 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → 𝑦 ∈ {𝑥 ∈ ω ∣ (𝐹𝑥) ∈ (inl “ 𝐴)})
117116, 1syl6eleqr 2209 . . . . . . . . . 10 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → 𝑦𝑆)
118117, 107jca 302 . . . . . . . . 9 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (𝑦𝑆 ∧ (inl‘𝑚) = (𝐹𝑦)))
119118ex 114 . . . . . . . 8 ((𝜑𝑚𝐴) → ((𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)) → (𝑦𝑆 ∧ (inl‘𝑚) = (𝐹𝑦))))
120119eximdv 1834 . . . . . . 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 2397 . . . . . 6 (∃𝑦𝑆 (inl‘𝑚) = (𝐹𝑦) ↔ ∃𝑦(𝑦𝑆 ∧ (inl‘𝑚) = (𝐹𝑦)))
123121, 122sylibr 133 . . . . 5 ((𝜑𝑚𝐴) → ∃𝑦𝑆 (inl‘𝑚) = (𝐹𝑦))
124 f1ocnvfv1 5644 . . . . . . . . . 10 ((inl:V–1-1-onto→({∅} × V) ∧ 𝑚 ∈ V) → (inl‘(inl‘𝑚)) = 𝑚)
1255, 108, 124mp2an 420 . . . . . . . . 9 (inl‘(inl‘𝑚)) = 𝑚
126 simpr 109 . . . . . . . . . 10 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → (inl‘𝑚) = (𝐹𝑦))
127126fveq2d 5391 . . . . . . . . 9 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → (inl‘(inl‘𝑚)) = (inl‘(𝐹𝑦)))
128125, 127syl5eqr 2162 . . . . . . . 8 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → 𝑚 = (inl‘(𝐹𝑦)))
12913fveq1i 5388 . . . . . . . . . 10 (𝐺𝑦) = ((inl ∘ 𝐹)‘𝑦)
13018ad2antrr 477 . . . . . . . . . . 11 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1o))
1313sseli 3061 . . . . . . . . . . . 12 (𝑦𝑆𝑦 ∈ ω)
132131adantl 273 . . . . . . . . . . 11 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → 𝑦 ∈ ω)
133 fvco3 5458 . . . . . . . . . . 11 ((𝐹:ω⟶(𝐴 ⊔ 1o) ∧ 𝑦 ∈ ω) → ((inl ∘ 𝐹)‘𝑦) = (inl‘(𝐹𝑦)))
134130, 132, 133syl2anc 406 . . . . . . . . . 10 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → ((inl ∘ 𝐹)‘𝑦) = (inl‘(𝐹𝑦)))
135129, 134syl5eq 2160 . . . . . . . . 9 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → (𝐺𝑦) = (inl‘(𝐹𝑦)))
136135adantr 272 . . . . . . . 8 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → (𝐺𝑦) = (inl‘(𝐹𝑦)))
137128, 136eqtr4d 2151 . . . . . . 7 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → 𝑚 = (𝐺𝑦))
138137ex 114 . . . . . 6 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → ((inl‘𝑚) = (𝐹𝑦) → 𝑚 = (𝐺𝑦)))
139138reximdva 2509 . . . . 5 ((𝜑𝑚𝐴) → (∃𝑦𝑆 (inl‘𝑚) = (𝐹𝑦) → ∃𝑦𝑆 𝑚 = (𝐺𝑦)))
140123, 139mpd 13 . . . 4 ((𝜑𝑚𝐴) → ∃𝑦𝑆 𝑚 = (𝐺𝑦))
141140ralrimiva 2480 . . 3 (𝜑 → ∀𝑚𝐴𝑦𝑆 𝑚 = (𝐺𝑦))
142 dffo3 5533 . . 3 (𝐺:𝑆onto𝐴 ↔ (𝐺:𝑆𝐴 ∧ ∀𝑚𝐴𝑦𝑆 𝑚 = (𝐺𝑦)))
14397, 141, 142sylanbrc 411 . 2 (𝜑𝐺:𝑆onto𝐴)
14453, 55bitr3i 185 . . . . . . 7 ((𝐹𝑛) ∈ (𝐴 ⊔ 1o) ↔ ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ (𝐹𝑛) ∈ (inr “ 1o)))
14551, 144sylib 121 . . . . . 6 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ (𝐹𝑛) ∈ (inr “ 1o)))
14640orim2i 733 . . . . . 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 803 . . . . 5 (DECID (𝐹𝑛) ∈ (inl “ 𝐴) ↔ ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ ¬ (𝐹𝑛) ∈ (inl “ 𝐴)))
149147, 148sylibr 133 . . . 4 ((𝜑𝑛 ∈ ω) → DECID (𝐹𝑛) ∈ (inl “ 𝐴))
150 ibar 297 . . . . . . 7 (𝑛 ∈ ω → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴))))
151150adantl 273 . . . . . 6 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴))))
152151, 64syl6bbr 197 . . . . 5 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ 𝑛𝑆))
153152dcbid 806 . . . 4 ((𝜑𝑛 ∈ ω) → (DECID (𝐹𝑛) ∈ (inl “ 𝐴) ↔ DECID 𝑛𝑆))
154149, 153mpbid 146 . . 3 ((𝜑𝑛 ∈ ω) → DECID 𝑛𝑆)
155154ralrimiva 2480 . 2 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
1564, 143, 1553jca 1144 1 (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑆))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  DECID wdc 802  w3a 945   = wceq 1314  wex 1451  wcel 1463  wral 2391  wrex 2392  {crab 2395  Vcvv 2658  cun 3037  cin 3038  wss 3039  c0 3331  {csn 3495  ωcom 4472   × cxp 4505  ccnv 4506  dom cdm 4507  ran crn 4508  cima 4510  ccom 4511  Fun wfun 5085   Fn wfn 5086  wf 5087  ontowfo 5089  1-1-ontowf1o 5090  cfv 5091  1oc1o 6272  cdju 6888  inlcinl 6896  inrcinr 6897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899
This theorem is referenced by:  ctssdclemr  6963  ctiunct  11848
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