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Theorem ctssdccl 7045
 Description: A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7047 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 3213 . . . 4 {𝑥 ∈ ω ∣ (𝐹𝑥) ∈ (inl “ 𝐴)} ⊆ ω
31, 2eqsstri 3160 . . 3 𝑆 ⊆ ω
43a1i 9 . 2 (𝜑𝑆 ⊆ ω)
5 djulf1o 6992 . . . . . . 7 inl:V–1-1-onto→({∅} × V)
6 f1ocnv 5424 . . . . . . 7 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
7 f1ofun 5413 . . . . . . 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 5390 . . . . . . 7 (𝐹:ω–onto→(𝐴 ⊔ 1o) → Fun 𝐹)
119, 10syl 14 . . . . . 6 (𝜑 → Fun 𝐹)
12 funco 5207 . . . . . . 7 ((Fun inl ∧ Fun 𝐹) → Fun (inl ∘ 𝐹))
13 ctssdccl.g . . . . . . . 8 𝐺 = (inl ∘ 𝐹)
1413funeqi 5188 . . . . . . 7 (Fun 𝐺 ↔ Fun (inl ∘ 𝐹))
1512, 14sylibr 133 . . . . . 6 ((Fun inl ∧ Fun 𝐹) → Fun 𝐺)
168, 11, 15sylancr 411 . . . . 5 (𝜑 → Fun 𝐺)
17 fof 5389 . . . . . . . . . . . 12 (𝐹:ω–onto→(𝐴 ⊔ 1o) → 𝐹:ω⟶(𝐴 ⊔ 1o))
189, 17syl 14 . . . . . . . . . . 11 (𝜑𝐹:ω⟶(𝐴 ⊔ 1o))
1918fdmd 5323 . . . . . . . . . 10 (𝜑 → dom 𝐹 = ω)
2019eleq2d 2227 . . . . . . . . 9 (𝜑 → (𝑛 ∈ dom 𝐹𝑛 ∈ ω))
2120anbi1d 461 . . . . . . . 8 (𝜑 → ((𝑛 ∈ dom 𝐹 ∧ (𝐹𝑛) ∈ dom inl) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ dom inl)))
22 dmcoss 4852 . . . . . . . . . . . 12 dom (inl ∘ 𝐹) ⊆ dom 𝐹
2322sseli 3124 . . . . . . . . . . 11 (𝑛 ∈ dom (inl ∘ 𝐹) → 𝑛 ∈ dom 𝐹)
2423pm4.71ri 390 . . . . . . . . . 10 (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹𝑛 ∈ dom (inl ∘ 𝐹)))
25 dmfco 5533 . . . . . . . . . . 11 ((Fun 𝐹𝑛 ∈ dom 𝐹) → (𝑛 ∈ dom (inl ∘ 𝐹) ↔ (𝐹𝑛) ∈ dom inl))
2625pm5.32da 448 . . . . . . . . . 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 4936 . . . . . . . . . . . . . 14 (inl “ 𝐴) ⊆ ran inl
3130sseli 3124 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (inl “ 𝐴) → (𝐹𝑛) ∈ ran inl)
3231adantl 275 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → (𝐹𝑛) ∈ ran inl)
33 df-rn 4594 . . . . . . . . . . . . 13 ran inl = dom inl
3433eleq2i 2224 . . . . . . . . . . . 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 6998 . . . . . . . . . . . . . 14 ((inl “ 𝐴) ∩ (inr “ 1o)) = ∅
38 disjel 3448 . . . . . . . . . . . . . 14 ((((inl “ 𝐴) ∩ (inr “ 1o)) = ∅ ∧ (𝐹𝑛) ∈ (inl “ 𝐴)) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
3937, 38mpan 421 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (inl “ 𝐴) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
4039con2i 617 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ (inr “ 1o) → ¬ (𝐹𝑛) ∈ (inl “ 𝐴))
4140adantl 275 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ¬ (𝐹𝑛) ∈ (inl “ 𝐴))
42 djuin 6998 . . . . . . . . . . . . . . . 16 ((inl “ V) ∩ (inr “ 1o)) = ∅
43 disjel 3448 . . . . . . . . . . . . . . . 16 ((((inl “ V) ∩ (inr “ 1o)) = ∅ ∧ (𝐹𝑛) ∈ (inl “ V)) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
4442, 43mpan 421 . . . . . . . . . . . . . . 15 ((𝐹𝑛) ∈ (inl “ V) → ¬ (𝐹𝑛) ∈ (inr “ 1o))
45 dfrn4 5043 . . . . . . . . . . . . . . 15 ran inl = (inl “ V)
4644, 45eleq2s 2252 . . . . . . . . . . . . . 14 ((𝐹𝑛) ∈ ran inl → ¬ (𝐹𝑛) ∈ (inr “ 1o))
4746con2i 617 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (inr “ 1o) → ¬ (𝐹𝑛) ∈ ran inl)
4847adantl 275 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ¬ (𝐹𝑛) ∈ ran inl)
4948, 34sylnib 666 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ¬ (𝐹𝑛) ∈ dom inl)
5041, 492falsed 692 . . . . . . . . . 10 (((𝜑𝑛 ∈ ω) ∧ (𝐹𝑛) ∈ (inr “ 1o)) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ dom inl))
5118ffvelrnda 5599 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ω) → (𝐹𝑛) ∈ (𝐴 ⊔ 1o))
52 djuun 7001 . . . . . . . . . . . . 13 ((inl “ 𝐴) ∪ (inr “ 1o)) = (𝐴 ⊔ 1o)
5352eleq2i 2224 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)) ↔ (𝐹𝑛) ∈ (𝐴 ⊔ 1o))
5451, 53sylibr 133 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ω) → (𝐹𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)))
55 elun 3248 . . . . . . . . . . 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 788 . . . . . . . . 9 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ dom inl))
5857pm5.32da 448 . . . . . . . 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 4784 . . . . . . . 8 dom 𝐺 = dom (inl ∘ 𝐹)
6160eleq2i 2224 . . . . . . 7 (𝑛 ∈ dom 𝐺𝑛 ∈ dom (inl ∘ 𝐹))
62 fveq2 5465 . . . . . . . . 9 (𝑥 = 𝑛 → (𝐹𝑥) = (𝐹𝑛))
6362eleq1d 2226 . . . . . . . 8 (𝑥 = 𝑛 → ((𝐹𝑥) ∈ (inl “ 𝐴) ↔ (𝐹𝑛) ∈ (inl “ 𝐴)))
6463, 1elrab2 2871 . . . . . . 7 (𝑛𝑆 ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴)))
6559, 61, 643bitr4g 222 . . . . . 6 (𝜑 → (𝑛 ∈ dom 𝐺𝑛𝑆))
6665eqrdv 2155 . . . . 5 (𝜑 → dom 𝐺 = 𝑆)
67 df-fn 5170 . . . . 5 (𝐺 Fn 𝑆 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝑆))
6816, 66, 67sylanbrc 414 . . . 4 (𝜑𝐺 Fn 𝑆)
6913fveq1i 5466 . . . . . . 7 (𝐺𝑚) = ((inl ∘ 𝐹)‘𝑚)
7018adantr 274 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1o))
71 fveq2 5465 . . . . . . . . . . . . 13 (𝑥 = 𝑚 → (𝐹𝑥) = (𝐹𝑚))
7271eleq1d 2226 . . . . . . . . . . . 12 (𝑥 = 𝑚 → ((𝐹𝑥) ∈ (inl “ 𝐴) ↔ (𝐹𝑚) ∈ (inl “ 𝐴)))
7372, 1elrab2 2871 . . . . . . . . . . 11 (𝑚𝑆 ↔ (𝑚 ∈ ω ∧ (𝐹𝑚) ∈ (inl “ 𝐴)))
7473biimpi 119 . . . . . . . . . 10 (𝑚𝑆 → (𝑚 ∈ ω ∧ (𝐹𝑚) ∈ (inl “ 𝐴)))
7574adantl 275 . . . . . . . . 9 ((𝜑𝑚𝑆) → (𝑚 ∈ ω ∧ (𝐹𝑚) ∈ (inl “ 𝐴)))
7675simpld 111 . . . . . . . 8 ((𝜑𝑚𝑆) → 𝑚 ∈ ω)
77 fvco3 5536 . . . . . . . 8 ((𝐹:ω⟶(𝐴 ⊔ 1o) ∧ 𝑚 ∈ ω) → ((inl ∘ 𝐹)‘𝑚) = (inl‘(𝐹𝑚)))
7870, 76, 77syl2anc 409 . . . . . . 7 ((𝜑𝑚𝑆) → ((inl ∘ 𝐹)‘𝑚) = (inl‘(𝐹𝑚)))
7969, 78syl5eq 2202 . . . . . 6 ((𝜑𝑚𝑆) → (𝐺𝑚) = (inl‘(𝐹𝑚)))
80 f1ofun 5413 . . . . . . . . . 10 (inl:V–1-1-onto→({∅} × V) → Fun inl)
815, 80ax-mp 5 . . . . . . . . 9 Fun inl
82 fvelima 5517 . . . . . . . . 9 ((Fun inl ∧ (𝐹𝑚) ∈ (inl “ 𝐴)) → ∃𝑧𝐴 (inl‘𝑧) = (𝐹𝑚))
8381, 82mpan 421 . . . . . . . 8 ((𝐹𝑚) ∈ (inl “ 𝐴) → ∃𝑧𝐴 (inl‘𝑧) = (𝐹𝑚))
8475, 83simpl2im 384 . . . . . . 7 ((𝜑𝑚𝑆) → ∃𝑧𝐴 (inl‘𝑧) = (𝐹𝑚))
85 simprr 522 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘𝑧) = (𝐹𝑚))
8685fveq2d 5469 . . . . . . . 8 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘(inl‘𝑧)) = (inl‘(𝐹𝑚)))
87 vex 2715 . . . . . . . . . 10 𝑧 ∈ V
88 f1ocnvfv1 5722 . . . . . . . . . 10 ((inl:V–1-1-onto→({∅} × V) ∧ 𝑧 ∈ V) → (inl‘(inl‘𝑧)) = 𝑧)
895, 87, 88mp2an 423 . . . . . . . . 9 (inl‘(inl‘𝑧)) = 𝑧
90 simprl 521 . . . . . . . . 9 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → 𝑧𝐴)
9189, 90eqeltrid 2244 . . . . . . . 8 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘(inl‘𝑧)) ∈ 𝐴)
9286, 91eqeltrrd 2235 . . . . . . 7 (((𝜑𝑚𝑆) ∧ (𝑧𝐴 ∧ (inl‘𝑧) = (𝐹𝑚))) → (inl‘(𝐹𝑚)) ∈ 𝐴)
9384, 92rexlimddv 2579 . . . . . 6 ((𝜑𝑚𝑆) → (inl‘(𝐹𝑚)) ∈ 𝐴)
9479, 93eqeltrd 2234 . . . . 5 ((𝜑𝑚𝑆) → (𝐺𝑚) ∈ 𝐴)
9594ralrimiva 2530 . . . 4 (𝜑 → ∀𝑚𝑆 (𝐺𝑚) ∈ 𝐴)
96 ffnfv 5622 . . . 4 (𝐺:𝑆𝐴 ↔ (𝐺 Fn 𝑆 ∧ ∀𝑚𝑆 (𝐺𝑚) ∈ 𝐴))
9768, 95, 96sylanbrc 414 . . 3 (𝜑𝐺:𝑆𝐴)
98 djulcl 6985 . . . . . . . 8 (𝑚𝐴 → (inl‘𝑚) ∈ (𝐴 ⊔ 1o))
99 foelrn 5698 . . . . . . . . . 10 ((𝐹:ω–onto→(𝐴 ⊔ 1o) ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹𝑦))
1009, 99sylan 281 . . . . . . . . 9 ((𝜑 ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹𝑦))
101 df-rex 2441 . . . . . . . . 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 284 . . . . . . 7 ((𝜑𝑚𝐴) → ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦)))
104 fveq2 5465 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
105104eleq1d 2226 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ (inl “ 𝐴) ↔ (𝐹𝑦) ∈ (inl “ 𝐴)))
106 simprl 521 . . . . . . . . . . . 12 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → 𝑦 ∈ ω)
107 simprr 522 . . . . . . . . . . . . 13 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (inl‘𝑚) = (𝐹𝑦))
108 vex 2715 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
109 f1odm 5415 . . . . . . . . . . . . . . . . 17 (inl:V–1-1-onto→({∅} × V) → dom inl = V)
1105, 109ax-mp 5 . . . . . . . . . . . . . . . 16 dom inl = V
111108, 110eleqtrri 2233 . . . . . . . . . . . . . . 15 𝑚 ∈ dom inl
112 funfvima 5693 . . . . . . . . . . . . . . 15 ((Fun inl ∧ 𝑚 ∈ dom inl) → (𝑚𝐴 → (inl‘𝑚) ∈ (inl “ 𝐴)))
11381, 111, 112mp2an 423 . . . . . . . . . . . . . 14 (𝑚𝐴 → (inl‘𝑚) ∈ (inl “ 𝐴))
114113ad2antlr 481 . . . . . . . . . . . . 13 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (inl‘𝑚) ∈ (inl “ 𝐴))
115107, 114eqeltrrd 2235 . . . . . . . . . . . 12 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → (𝐹𝑦) ∈ (inl “ 𝐴))
116105, 106, 115elrabd 2870 . . . . . . . . . . 11 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → 𝑦 ∈ {𝑥 ∈ ω ∣ (𝐹𝑥) ∈ (inl “ 𝐴)})
117116, 1eleqtrrdi 2251 . . . . . . . . . 10 (((𝜑𝑚𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹𝑦))) → 𝑦𝑆)
118117, 107jca 304 . . . . . . . . 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 1860 . . . . . . 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 2441 . . . . . 6 (∃𝑦𝑆 (inl‘𝑚) = (𝐹𝑦) ↔ ∃𝑦(𝑦𝑆 ∧ (inl‘𝑚) = (𝐹𝑦)))
123121, 122sylibr 133 . . . . 5 ((𝜑𝑚𝐴) → ∃𝑦𝑆 (inl‘𝑚) = (𝐹𝑦))
124 f1ocnvfv1 5722 . . . . . . . . . 10 ((inl:V–1-1-onto→({∅} × V) ∧ 𝑚 ∈ V) → (inl‘(inl‘𝑚)) = 𝑚)
1255, 108, 124mp2an 423 . . . . . . . . 9 (inl‘(inl‘𝑚)) = 𝑚
126 simpr 109 . . . . . . . . . 10 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → (inl‘𝑚) = (𝐹𝑦))
127126fveq2d 5469 . . . . . . . . 9 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → (inl‘(inl‘𝑚)) = (inl‘(𝐹𝑦)))
128125, 127syl5eqr 2204 . . . . . . . 8 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → 𝑚 = (inl‘(𝐹𝑦)))
12913fveq1i 5466 . . . . . . . . . 10 (𝐺𝑦) = ((inl ∘ 𝐹)‘𝑦)
13018ad2antrr 480 . . . . . . . . . . 11 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1o))
1313sseli 3124 . . . . . . . . . . . 12 (𝑦𝑆𝑦 ∈ ω)
132131adantl 275 . . . . . . . . . . 11 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → 𝑦 ∈ ω)
133 fvco3 5536 . . . . . . . . . . 11 ((𝐹:ω⟶(𝐴 ⊔ 1o) ∧ 𝑦 ∈ ω) → ((inl ∘ 𝐹)‘𝑦) = (inl‘(𝐹𝑦)))
134130, 132, 133syl2anc 409 . . . . . . . . . 10 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → ((inl ∘ 𝐹)‘𝑦) = (inl‘(𝐹𝑦)))
135129, 134syl5eq 2202 . . . . . . . . 9 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → (𝐺𝑦) = (inl‘(𝐹𝑦)))
136135adantr 274 . . . . . . . 8 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → (𝐺𝑦) = (inl‘(𝐹𝑦)))
137128, 136eqtr4d 2193 . . . . . . 7 ((((𝜑𝑚𝐴) ∧ 𝑦𝑆) ∧ (inl‘𝑚) = (𝐹𝑦)) → 𝑚 = (𝐺𝑦))
138137ex 114 . . . . . 6 (((𝜑𝑚𝐴) ∧ 𝑦𝑆) → ((inl‘𝑚) = (𝐹𝑦) → 𝑚 = (𝐺𝑦)))
139138reximdva 2559 . . . . 5 ((𝜑𝑚𝐴) → (∃𝑦𝑆 (inl‘𝑚) = (𝐹𝑦) → ∃𝑦𝑆 𝑚 = (𝐺𝑦)))
140123, 139mpd 13 . . . 4 ((𝜑𝑚𝐴) → ∃𝑦𝑆 𝑚 = (𝐺𝑦))
141140ralrimiva 2530 . . 3 (𝜑 → ∀𝑚𝐴𝑦𝑆 𝑚 = (𝐺𝑦))
142 dffo3 5611 . . 3 (𝐺:𝑆onto𝐴 ↔ (𝐺:𝑆𝐴 ∧ ∀𝑚𝐴𝑦𝑆 𝑚 = (𝐺𝑦)))
14397, 141, 142sylanbrc 414 . 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 751 . . . . . 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 821 . . . . 5 (DECID (𝐹𝑛) ∈ (inl “ 𝐴) ↔ ((𝐹𝑛) ∈ (inl “ 𝐴) ∨ ¬ (𝐹𝑛) ∈ (inl “ 𝐴)))
149147, 148sylibr 133 . . . 4 ((𝜑𝑛 ∈ ω) → DECID (𝐹𝑛) ∈ (inl “ 𝐴))
150 ibar 299 . . . . . . 7 (𝑛 ∈ ω → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴))))
151150adantl 275 . . . . . 6 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ (𝑛 ∈ ω ∧ (𝐹𝑛) ∈ (inl “ 𝐴))))
152151, 64bitr4di 197 . . . . 5 ((𝜑𝑛 ∈ ω) → ((𝐹𝑛) ∈ (inl “ 𝐴) ↔ 𝑛𝑆))
153152dcbid 824 . . . 4 ((𝜑𝑛 ∈ ω) → (DECID (𝐹𝑛) ∈ (inl “ 𝐴) ↔ DECID 𝑛𝑆))
154149, 153mpbid 146 . . 3 ((𝜑𝑛 ∈ ω) → DECID 𝑛𝑆)
155154ralrimiva 2530 . 2 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
1564, 143, 1553jca 1162 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 698  DECID wdc 820   ∧ w3a 963   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436  {crab 2439  Vcvv 2712   ∪ cun 3100   ∩ cin 3101   ⊆ wss 3102  ∅c0 3394  {csn 3560  ωcom 4547   × cxp 4581  ◡ccnv 4582  dom cdm 4583  ran crn 4584   “ cima 4586   ∘ ccom 4587  Fun wfun 5161   Fn wfn 5162  ⟶wf 5163  –onto→wfo 5165  –1-1-onto→wf1o 5166  ‘cfv 5167  1oc1o 6350   ⊔ cdju 6971  inlcinl 6979  inrcinr 6980 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-1st 6082  df-2nd 6083  df-1o 6357  df-dju 6972  df-inl 6981  df-inr 6982 This theorem is referenced by:  ctssdclemr  7046  ctiunct  12141
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