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Theorem ctssdccl 7109
Description: A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7111 but expressed in terms of classes rather than βˆƒ. (Contributed by Jim Kingdon, 30-Oct-2023.)
Hypotheses
Ref Expression
ctssdccl.f (πœ‘ β†’ 𝐹:ω–ontoβ†’(𝐴 βŠ” 1o))
ctssdccl.s 𝑆 = {π‘₯ ∈ Ο‰ ∣ (πΉβ€˜π‘₯) ∈ (inl β€œ 𝐴)}
ctssdccl.g 𝐺 = (β—‘inl ∘ 𝐹)
Assertion
Ref Expression
ctssdccl (πœ‘ β†’ (𝑆 βŠ† Ο‰ ∧ 𝐺:𝑆–onto→𝐴 ∧ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆))
Distinct variable groups:   π‘₯,𝐴   𝑛,𝐹,π‘₯   𝑛,𝐺   𝑆,𝑛   πœ‘,𝑛
Allowed substitution hints:   πœ‘(π‘₯)   𝐴(𝑛)   𝑆(π‘₯)   𝐺(π‘₯)

Proof of Theorem ctssdccl
Dummy variables π‘š 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ctssdccl.s . . . 4 𝑆 = {π‘₯ ∈ Ο‰ ∣ (πΉβ€˜π‘₯) ∈ (inl β€œ 𝐴)}
2 ssrab2 3240 . . . 4 {π‘₯ ∈ Ο‰ ∣ (πΉβ€˜π‘₯) ∈ (inl β€œ 𝐴)} βŠ† Ο‰
31, 2eqsstri 3187 . . 3 𝑆 βŠ† Ο‰
43a1i 9 . 2 (πœ‘ β†’ 𝑆 βŠ† Ο‰)
5 djulf1o 7056 . . . . . . 7 inl:V–1-1-ontoβ†’({βˆ…} Γ— V)
6 f1ocnv 5474 . . . . . . 7 (inl:V–1-1-ontoβ†’({βˆ…} Γ— V) β†’ β—‘inl:({βˆ…} Γ— V)–1-1-ontoβ†’V)
7 f1ofun 5463 . . . . . . 7 (β—‘inl:({βˆ…} Γ— V)–1-1-ontoβ†’V β†’ Fun β—‘inl)
85, 6, 7mp2b 8 . . . . . 6 Fun β—‘inl
9 ctssdccl.f . . . . . . 7 (πœ‘ β†’ 𝐹:ω–ontoβ†’(𝐴 βŠ” 1o))
10 fofun 5439 . . . . . . 7 (𝐹:ω–ontoβ†’(𝐴 βŠ” 1o) β†’ Fun 𝐹)
119, 10syl 14 . . . . . 6 (πœ‘ β†’ Fun 𝐹)
12 funco 5256 . . . . . . 7 ((Fun β—‘inl ∧ Fun 𝐹) β†’ Fun (β—‘inl ∘ 𝐹))
13 ctssdccl.g . . . . . . . 8 𝐺 = (β—‘inl ∘ 𝐹)
1413funeqi 5237 . . . . . . 7 (Fun 𝐺 ↔ Fun (β—‘inl ∘ 𝐹))
1512, 14sylibr 134 . . . . . 6 ((Fun β—‘inl ∧ Fun 𝐹) β†’ Fun 𝐺)
168, 11, 15sylancr 414 . . . . 5 (πœ‘ β†’ Fun 𝐺)
17 fof 5438 . . . . . . . . . . . 12 (𝐹:ω–ontoβ†’(𝐴 βŠ” 1o) β†’ 𝐹:Ο‰βŸΆ(𝐴 βŠ” 1o))
189, 17syl 14 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹:Ο‰βŸΆ(𝐴 βŠ” 1o))
1918fdmd 5372 . . . . . . . . . 10 (πœ‘ β†’ dom 𝐹 = Ο‰)
2019eleq2d 2247 . . . . . . . . 9 (πœ‘ β†’ (𝑛 ∈ dom 𝐹 ↔ 𝑛 ∈ Ο‰))
2120anbi1d 465 . . . . . . . 8 (πœ‘ β†’ ((𝑛 ∈ dom 𝐹 ∧ (πΉβ€˜π‘›) ∈ dom β—‘inl) ↔ (𝑛 ∈ Ο‰ ∧ (πΉβ€˜π‘›) ∈ dom β—‘inl)))
22 dmcoss 4896 . . . . . . . . . . . 12 dom (β—‘inl ∘ 𝐹) βŠ† dom 𝐹
2322sseli 3151 . . . . . . . . . . 11 (𝑛 ∈ dom (β—‘inl ∘ 𝐹) β†’ 𝑛 ∈ dom 𝐹)
2423pm4.71ri 392 . . . . . . . . . 10 (𝑛 ∈ dom (β—‘inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹 ∧ 𝑛 ∈ dom (β—‘inl ∘ 𝐹)))
25 dmfco 5584 . . . . . . . . . . 11 ((Fun 𝐹 ∧ 𝑛 ∈ dom 𝐹) β†’ (𝑛 ∈ dom (β—‘inl ∘ 𝐹) ↔ (πΉβ€˜π‘›) ∈ dom β—‘inl))
2625pm5.32da 452 . . . . . . . . . 10 (Fun 𝐹 β†’ ((𝑛 ∈ dom 𝐹 ∧ 𝑛 ∈ dom (β—‘inl ∘ 𝐹)) ↔ (𝑛 ∈ dom 𝐹 ∧ (πΉβ€˜π‘›) ∈ dom β—‘inl)))
2724, 26bitrid 192 . . . . . . . . 9 (Fun 𝐹 β†’ (𝑛 ∈ dom (β—‘inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹 ∧ (πΉβ€˜π‘›) ∈ dom β—‘inl)))
2811, 27syl 14 . . . . . . . 8 (πœ‘ β†’ (𝑛 ∈ dom (β—‘inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹 ∧ (πΉβ€˜π‘›) ∈ dom β—‘inl)))
29 simpr 110 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)) β†’ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴))
30 imassrn 4981 . . . . . . . . . . . . . 14 (inl β€œ 𝐴) βŠ† ran inl
3130sseli 3151 . . . . . . . . . . . . 13 ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) β†’ (πΉβ€˜π‘›) ∈ ran inl)
3231adantl 277 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)) β†’ (πΉβ€˜π‘›) ∈ ran inl)
33 df-rn 4637 . . . . . . . . . . . . 13 ran inl = dom β—‘inl
3433eleq2i 2244 . . . . . . . . . . . 12 ((πΉβ€˜π‘›) ∈ ran inl ↔ (πΉβ€˜π‘›) ∈ dom β—‘inl)
3532, 34sylib 122 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)) β†’ (πΉβ€˜π‘›) ∈ dom β—‘inl)
3629, 352thd 175 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)) β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ↔ (πΉβ€˜π‘›) ∈ dom β—‘inl))
37 djuin 7062 . . . . . . . . . . . . . 14 ((inl β€œ 𝐴) ∩ (inr β€œ 1o)) = βˆ…
38 disjel 3477 . . . . . . . . . . . . . 14 ((((inl β€œ 𝐴) ∩ (inr β€œ 1o)) = βˆ… ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)) β†’ Β¬ (πΉβ€˜π‘›) ∈ (inr β€œ 1o))
3937, 38mpan 424 . . . . . . . . . . . . 13 ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) β†’ Β¬ (πΉβ€˜π‘›) ∈ (inr β€œ 1o))
4039con2i 627 . . . . . . . . . . . 12 ((πΉβ€˜π‘›) ∈ (inr β€œ 1o) β†’ Β¬ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴))
4140adantl 277 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) ∈ (inr β€œ 1o)) β†’ Β¬ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴))
42 djuin 7062 . . . . . . . . . . . . . . . 16 ((inl β€œ V) ∩ (inr β€œ 1o)) = βˆ…
43 disjel 3477 . . . . . . . . . . . . . . . 16 ((((inl β€œ V) ∩ (inr β€œ 1o)) = βˆ… ∧ (πΉβ€˜π‘›) ∈ (inl β€œ V)) β†’ Β¬ (πΉβ€˜π‘›) ∈ (inr β€œ 1o))
4442, 43mpan 424 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘›) ∈ (inl β€œ V) β†’ Β¬ (πΉβ€˜π‘›) ∈ (inr β€œ 1o))
45 dfrn4 5089 . . . . . . . . . . . . . . 15 ran inl = (inl β€œ V)
4644, 45eleq2s 2272 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘›) ∈ ran inl β†’ Β¬ (πΉβ€˜π‘›) ∈ (inr β€œ 1o))
4746con2i 627 . . . . . . . . . . . . 13 ((πΉβ€˜π‘›) ∈ (inr β€œ 1o) β†’ Β¬ (πΉβ€˜π‘›) ∈ ran inl)
4847adantl 277 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) ∈ (inr β€œ 1o)) β†’ Β¬ (πΉβ€˜π‘›) ∈ ran inl)
4948, 34sylnib 676 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) ∈ (inr β€œ 1o)) β†’ Β¬ (πΉβ€˜π‘›) ∈ dom β—‘inl)
5041, 492falsed 702 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) ∈ (inr β€œ 1o)) β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ↔ (πΉβ€˜π‘›) ∈ dom β—‘inl))
5118ffvelcdmda 5651 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ (πΉβ€˜π‘›) ∈ (𝐴 βŠ” 1o))
52 djuun 7065 . . . . . . . . . . . . 13 ((inl β€œ 𝐴) βˆͺ (inr β€œ 1o)) = (𝐴 βŠ” 1o)
5352eleq2i 2244 . . . . . . . . . . . 12 ((πΉβ€˜π‘›) ∈ ((inl β€œ 𝐴) βˆͺ (inr β€œ 1o)) ↔ (πΉβ€˜π‘›) ∈ (𝐴 βŠ” 1o))
5451, 53sylibr 134 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ (πΉβ€˜π‘›) ∈ ((inl β€œ 𝐴) βˆͺ (inr β€œ 1o)))
55 elun 3276 . . . . . . . . . . 11 ((πΉβ€˜π‘›) ∈ ((inl β€œ 𝐴) βˆͺ (inr β€œ 1o)) ↔ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ∨ (πΉβ€˜π‘›) ∈ (inr β€œ 1o)))
5654, 55sylib 122 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ∨ (πΉβ€˜π‘›) ∈ (inr β€œ 1o)))
5736, 50, 56mpjaodan 798 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ↔ (πΉβ€˜π‘›) ∈ dom β—‘inl))
5857pm5.32da 452 . . . . . . . 8 (πœ‘ β†’ ((𝑛 ∈ Ο‰ ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)) ↔ (𝑛 ∈ Ο‰ ∧ (πΉβ€˜π‘›) ∈ dom β—‘inl)))
5921, 28, 583bitr4d 220 . . . . . . 7 (πœ‘ β†’ (𝑛 ∈ dom (β—‘inl ∘ 𝐹) ↔ (𝑛 ∈ Ο‰ ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴))))
6013dmeqi 4828 . . . . . . . 8 dom 𝐺 = dom (β—‘inl ∘ 𝐹)
6160eleq2i 2244 . . . . . . 7 (𝑛 ∈ dom 𝐺 ↔ 𝑛 ∈ dom (β—‘inl ∘ 𝐹))
62 fveq2 5515 . . . . . . . . 9 (π‘₯ = 𝑛 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘›))
6362eleq1d 2246 . . . . . . . 8 (π‘₯ = 𝑛 β†’ ((πΉβ€˜π‘₯) ∈ (inl β€œ 𝐴) ↔ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)))
6463, 1elrab2 2896 . . . . . . 7 (𝑛 ∈ 𝑆 ↔ (𝑛 ∈ Ο‰ ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)))
6559, 61, 643bitr4g 223 . . . . . 6 (πœ‘ β†’ (𝑛 ∈ dom 𝐺 ↔ 𝑛 ∈ 𝑆))
6665eqrdv 2175 . . . . 5 (πœ‘ β†’ dom 𝐺 = 𝑆)
67 df-fn 5219 . . . . 5 (𝐺 Fn 𝑆 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝑆))
6816, 66, 67sylanbrc 417 . . . 4 (πœ‘ β†’ 𝐺 Fn 𝑆)
6913fveq1i 5516 . . . . . . 7 (πΊβ€˜π‘š) = ((β—‘inl ∘ 𝐹)β€˜π‘š)
7018adantr 276 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ 𝑆) β†’ 𝐹:Ο‰βŸΆ(𝐴 βŠ” 1o))
71 fveq2 5515 . . . . . . . . . . . . 13 (π‘₯ = π‘š β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘š))
7271eleq1d 2246 . . . . . . . . . . . 12 (π‘₯ = π‘š β†’ ((πΉβ€˜π‘₯) ∈ (inl β€œ 𝐴) ↔ (πΉβ€˜π‘š) ∈ (inl β€œ 𝐴)))
7372, 1elrab2 2896 . . . . . . . . . . 11 (π‘š ∈ 𝑆 ↔ (π‘š ∈ Ο‰ ∧ (πΉβ€˜π‘š) ∈ (inl β€œ 𝐴)))
7473biimpi 120 . . . . . . . . . 10 (π‘š ∈ 𝑆 β†’ (π‘š ∈ Ο‰ ∧ (πΉβ€˜π‘š) ∈ (inl β€œ 𝐴)))
7574adantl 277 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ 𝑆) β†’ (π‘š ∈ Ο‰ ∧ (πΉβ€˜π‘š) ∈ (inl β€œ 𝐴)))
7675simpld 112 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ 𝑆) β†’ π‘š ∈ Ο‰)
77 fvco3 5587 . . . . . . . 8 ((𝐹:Ο‰βŸΆ(𝐴 βŠ” 1o) ∧ π‘š ∈ Ο‰) β†’ ((β—‘inl ∘ 𝐹)β€˜π‘š) = (β—‘inlβ€˜(πΉβ€˜π‘š)))
7870, 76, 77syl2anc 411 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ 𝑆) β†’ ((β—‘inl ∘ 𝐹)β€˜π‘š) = (β—‘inlβ€˜(πΉβ€˜π‘š)))
7969, 78eqtrid 2222 . . . . . 6 ((πœ‘ ∧ π‘š ∈ 𝑆) β†’ (πΊβ€˜π‘š) = (β—‘inlβ€˜(πΉβ€˜π‘š)))
80 f1ofun 5463 . . . . . . . . . 10 (inl:V–1-1-ontoβ†’({βˆ…} Γ— V) β†’ Fun inl)
815, 80ax-mp 5 . . . . . . . . 9 Fun inl
82 fvelima 5567 . . . . . . . . 9 ((Fun inl ∧ (πΉβ€˜π‘š) ∈ (inl β€œ 𝐴)) β†’ βˆƒπ‘§ ∈ 𝐴 (inlβ€˜π‘§) = (πΉβ€˜π‘š))
8381, 82mpan 424 . . . . . . . 8 ((πΉβ€˜π‘š) ∈ (inl β€œ 𝐴) β†’ βˆƒπ‘§ ∈ 𝐴 (inlβ€˜π‘§) = (πΉβ€˜π‘š))
8475, 83simpl2im 386 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ 𝑆) β†’ βˆƒπ‘§ ∈ 𝐴 (inlβ€˜π‘§) = (πΉβ€˜π‘š))
85 simprr 531 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inlβ€˜π‘§) = (πΉβ€˜π‘š))) β†’ (inlβ€˜π‘§) = (πΉβ€˜π‘š))
8685fveq2d 5519 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inlβ€˜π‘§) = (πΉβ€˜π‘š))) β†’ (β—‘inlβ€˜(inlβ€˜π‘§)) = (β—‘inlβ€˜(πΉβ€˜π‘š)))
87 vex 2740 . . . . . . . . . 10 𝑧 ∈ V
88 f1ocnvfv1 5777 . . . . . . . . . 10 ((inl:V–1-1-ontoβ†’({βˆ…} Γ— V) ∧ 𝑧 ∈ V) β†’ (β—‘inlβ€˜(inlβ€˜π‘§)) = 𝑧)
895, 87, 88mp2an 426 . . . . . . . . 9 (β—‘inlβ€˜(inlβ€˜π‘§)) = 𝑧
90 simprl 529 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inlβ€˜π‘§) = (πΉβ€˜π‘š))) β†’ 𝑧 ∈ 𝐴)
9189, 90eqeltrid 2264 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inlβ€˜π‘§) = (πΉβ€˜π‘š))) β†’ (β—‘inlβ€˜(inlβ€˜π‘§)) ∈ 𝐴)
9286, 91eqeltrrd 2255 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inlβ€˜π‘§) = (πΉβ€˜π‘š))) β†’ (β—‘inlβ€˜(πΉβ€˜π‘š)) ∈ 𝐴)
9384, 92rexlimddv 2599 . . . . . 6 ((πœ‘ ∧ π‘š ∈ 𝑆) β†’ (β—‘inlβ€˜(πΉβ€˜π‘š)) ∈ 𝐴)
9479, 93eqeltrd 2254 . . . . 5 ((πœ‘ ∧ π‘š ∈ 𝑆) β†’ (πΊβ€˜π‘š) ∈ 𝐴)
9594ralrimiva 2550 . . . 4 (πœ‘ β†’ βˆ€π‘š ∈ 𝑆 (πΊβ€˜π‘š) ∈ 𝐴)
96 ffnfv 5674 . . . 4 (𝐺:π‘†βŸΆπ΄ ↔ (𝐺 Fn 𝑆 ∧ βˆ€π‘š ∈ 𝑆 (πΊβ€˜π‘š) ∈ 𝐴))
9768, 95, 96sylanbrc 417 . . 3 (πœ‘ β†’ 𝐺:π‘†βŸΆπ΄)
98 djulcl 7049 . . . . . . . 8 (π‘š ∈ 𝐴 β†’ (inlβ€˜π‘š) ∈ (𝐴 βŠ” 1o))
99 foelrn 5753 . . . . . . . . . 10 ((𝐹:ω–ontoβ†’(𝐴 βŠ” 1o) ∧ (inlβ€˜π‘š) ∈ (𝐴 βŠ” 1o)) β†’ βˆƒπ‘¦ ∈ Ο‰ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))
1009, 99sylan 283 . . . . . . . . 9 ((πœ‘ ∧ (inlβ€˜π‘š) ∈ (𝐴 βŠ” 1o)) β†’ βˆƒπ‘¦ ∈ Ο‰ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))
101 df-rex 2461 . . . . . . . . 9 (βˆƒπ‘¦ ∈ Ο‰ (inlβ€˜π‘š) = (πΉβ€˜π‘¦) ↔ βˆƒπ‘¦(𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)))
102100, 101sylib 122 . . . . . . . 8 ((πœ‘ ∧ (inlβ€˜π‘š) ∈ (𝐴 βŠ” 1o)) β†’ βˆƒπ‘¦(𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)))
10398, 102sylan2 286 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ 𝐴) β†’ βˆƒπ‘¦(𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)))
104 fveq2 5515 . . . . . . . . . . . . 13 (π‘₯ = 𝑦 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
105104eleq1d 2246 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ ((πΉβ€˜π‘₯) ∈ (inl β€œ 𝐴) ↔ (πΉβ€˜π‘¦) ∈ (inl β€œ 𝐴)))
106 simprl 529 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ (𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))) β†’ 𝑦 ∈ Ο‰)
107 simprr 531 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ (𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))) β†’ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))
108 vex 2740 . . . . . . . . . . . . . . . 16 π‘š ∈ V
109 f1odm 5465 . . . . . . . . . . . . . . . . 17 (inl:V–1-1-ontoβ†’({βˆ…} Γ— V) β†’ dom inl = V)
1105, 109ax-mp 5 . . . . . . . . . . . . . . . 16 dom inl = V
111108, 110eleqtrri 2253 . . . . . . . . . . . . . . 15 π‘š ∈ dom inl
112 funfvima 5748 . . . . . . . . . . . . . . 15 ((Fun inl ∧ π‘š ∈ dom inl) β†’ (π‘š ∈ 𝐴 β†’ (inlβ€˜π‘š) ∈ (inl β€œ 𝐴)))
11381, 111, 112mp2an 426 . . . . . . . . . . . . . 14 (π‘š ∈ 𝐴 β†’ (inlβ€˜π‘š) ∈ (inl β€œ 𝐴))
114113ad2antlr 489 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ (𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))) β†’ (inlβ€˜π‘š) ∈ (inl β€œ 𝐴))
115107, 114eqeltrrd 2255 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ (𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))) β†’ (πΉβ€˜π‘¦) ∈ (inl β€œ 𝐴))
116105, 106, 115elrabd 2895 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ (𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))) β†’ 𝑦 ∈ {π‘₯ ∈ Ο‰ ∣ (πΉβ€˜π‘₯) ∈ (inl β€œ 𝐴)})
117116, 1eleqtrrdi 2271 . . . . . . . . . 10 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ (𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))) β†’ 𝑦 ∈ 𝑆)
118117, 107jca 306 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ (𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))) β†’ (𝑦 ∈ 𝑆 ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)))
119118ex 115 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ 𝐴) β†’ ((𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)) β†’ (𝑦 ∈ 𝑆 ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))))
120119eximdv 1880 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ 𝐴) β†’ (βˆƒπ‘¦(𝑦 ∈ Ο‰ ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)) β†’ βˆƒπ‘¦(𝑦 ∈ 𝑆 ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))))
121103, 120mpd 13 . . . . . 6 ((πœ‘ ∧ π‘š ∈ 𝐴) β†’ βˆƒπ‘¦(𝑦 ∈ 𝑆 ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)))
122 df-rex 2461 . . . . . 6 (βˆƒπ‘¦ ∈ 𝑆 (inlβ€˜π‘š) = (πΉβ€˜π‘¦) ↔ βˆƒπ‘¦(𝑦 ∈ 𝑆 ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)))
123121, 122sylibr 134 . . . . 5 ((πœ‘ ∧ π‘š ∈ 𝐴) β†’ βˆƒπ‘¦ ∈ 𝑆 (inlβ€˜π‘š) = (πΉβ€˜π‘¦))
124 f1ocnvfv1 5777 . . . . . . . . . 10 ((inl:V–1-1-ontoβ†’({βˆ…} Γ— V) ∧ π‘š ∈ V) β†’ (β—‘inlβ€˜(inlβ€˜π‘š)) = π‘š)
1255, 108, 124mp2an 426 . . . . . . . . 9 (β—‘inlβ€˜(inlβ€˜π‘š)) = π‘š
126 simpr 110 . . . . . . . . . 10 ((((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)) β†’ (inlβ€˜π‘š) = (πΉβ€˜π‘¦))
127126fveq2d 5519 . . . . . . . . 9 ((((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)) β†’ (β—‘inlβ€˜(inlβ€˜π‘š)) = (β—‘inlβ€˜(πΉβ€˜π‘¦)))
128125, 127eqtr3id 2224 . . . . . . . 8 ((((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)) β†’ π‘š = (β—‘inlβ€˜(πΉβ€˜π‘¦)))
12913fveq1i 5516 . . . . . . . . . 10 (πΊβ€˜π‘¦) = ((β—‘inl ∘ 𝐹)β€˜π‘¦)
13018ad2antrr 488 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) β†’ 𝐹:Ο‰βŸΆ(𝐴 βŠ” 1o))
1313sseli 3151 . . . . . . . . . . . 12 (𝑦 ∈ 𝑆 β†’ 𝑦 ∈ Ο‰)
132131adantl 277 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) β†’ 𝑦 ∈ Ο‰)
133 fvco3 5587 . . . . . . . . . . 11 ((𝐹:Ο‰βŸΆ(𝐴 βŠ” 1o) ∧ 𝑦 ∈ Ο‰) β†’ ((β—‘inl ∘ 𝐹)β€˜π‘¦) = (β—‘inlβ€˜(πΉβ€˜π‘¦)))
134130, 132, 133syl2anc 411 . . . . . . . . . 10 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) β†’ ((β—‘inl ∘ 𝐹)β€˜π‘¦) = (β—‘inlβ€˜(πΉβ€˜π‘¦)))
135129, 134eqtrid 2222 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) β†’ (πΊβ€˜π‘¦) = (β—‘inlβ€˜(πΉβ€˜π‘¦)))
136135adantr 276 . . . . . . . 8 ((((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)) β†’ (πΊβ€˜π‘¦) = (β—‘inlβ€˜(πΉβ€˜π‘¦)))
137128, 136eqtr4d 2213 . . . . . . 7 ((((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inlβ€˜π‘š) = (πΉβ€˜π‘¦)) β†’ π‘š = (πΊβ€˜π‘¦))
138137ex 115 . . . . . 6 (((πœ‘ ∧ π‘š ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) β†’ ((inlβ€˜π‘š) = (πΉβ€˜π‘¦) β†’ π‘š = (πΊβ€˜π‘¦)))
139138reximdva 2579 . . . . 5 ((πœ‘ ∧ π‘š ∈ 𝐴) β†’ (βˆƒπ‘¦ ∈ 𝑆 (inlβ€˜π‘š) = (πΉβ€˜π‘¦) β†’ βˆƒπ‘¦ ∈ 𝑆 π‘š = (πΊβ€˜π‘¦)))
140123, 139mpd 13 . . . 4 ((πœ‘ ∧ π‘š ∈ 𝐴) β†’ βˆƒπ‘¦ ∈ 𝑆 π‘š = (πΊβ€˜π‘¦))
141140ralrimiva 2550 . . 3 (πœ‘ β†’ βˆ€π‘š ∈ 𝐴 βˆƒπ‘¦ ∈ 𝑆 π‘š = (πΊβ€˜π‘¦))
142 dffo3 5663 . . 3 (𝐺:𝑆–onto→𝐴 ↔ (𝐺:π‘†βŸΆπ΄ ∧ βˆ€π‘š ∈ 𝐴 βˆƒπ‘¦ ∈ 𝑆 π‘š = (πΊβ€˜π‘¦)))
14397, 141, 142sylanbrc 417 . 2 (πœ‘ β†’ 𝐺:𝑆–onto→𝐴)
14453, 55bitr3i 186 . . . . . . 7 ((πΉβ€˜π‘›) ∈ (𝐴 βŠ” 1o) ↔ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ∨ (πΉβ€˜π‘›) ∈ (inr β€œ 1o)))
14551, 144sylib 122 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ∨ (πΉβ€˜π‘›) ∈ (inr β€œ 1o)))
14640orim2i 761 . . . . . 6 (((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ∨ (πΉβ€˜π‘›) ∈ (inr β€œ 1o)) β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ∨ Β¬ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)))
147145, 146syl 14 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ∨ Β¬ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)))
148 df-dc 835 . . . . 5 (DECID (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ↔ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ∨ Β¬ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴)))
149147, 148sylibr 134 . . . 4 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ DECID (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴))
150 ibar 301 . . . . . . 7 (𝑛 ∈ Ο‰ β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ↔ (𝑛 ∈ Ο‰ ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴))))
151150adantl 277 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ↔ (𝑛 ∈ Ο‰ ∧ (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴))))
152151, 64bitr4di 198 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ↔ 𝑛 ∈ 𝑆))
153152dcbid 838 . . . 4 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ (DECID (πΉβ€˜π‘›) ∈ (inl β€œ 𝐴) ↔ DECID 𝑛 ∈ 𝑆))
154149, 153mpbid 147 . . 3 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ DECID 𝑛 ∈ 𝑆)
155154ralrimiva 2550 . 2 (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)
1564, 143, 1553jca 1177 1 (πœ‘ β†’ (𝑆 βŠ† Ο‰ ∧ 𝐺:𝑆–onto→𝐴 ∧ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆))
Colors of variables: wff set class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   ∧ w3a 978   = wceq 1353  βˆƒwex 1492   ∈ wcel 2148  βˆ€wral 2455  βˆƒwrex 2456  {crab 2459  Vcvv 2737   βˆͺ cun 3127   ∩ cin 3128   βŠ† wss 3129  βˆ…c0 3422  {csn 3592  Ο‰com 4589   Γ— cxp 4624  β—‘ccnv 4625  dom cdm 4626  ran crn 4627   β€œ cima 4629   ∘ ccom 4630  Fun wfun 5210   Fn wfn 5211  βŸΆwf 5212  β€“ontoβ†’wfo 5214  β€“1-1-ontoβ†’wf1o 5215  β€˜cfv 5216  1oc1o 6409   βŠ” cdju 7035  inlcinl 7043  inrcinr 7044
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 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-dju 7036  df-inl 7045  df-inr 7046
This theorem is referenced by:  ctssdclemr  7110  ctiunct  12440
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