ctssdccl - Intuitionistic Logic Explorer

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Theorem ctssdccl 7124

Description: A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7126 but expressed in terms of classes rather than ∃. (Contributed by Jim Kingdon, 30-Oct-2023.)

**Hypotheses**
Ref	Expression
ctssdccl.f	⊢ (𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1_o))
ctssdccl.s	⊢ 𝑆 = {𝑥 ∈ ω ∣ (𝐹‘𝑥) ∈ (inl “ 𝐴)}
ctssdccl.g	⊢ 𝐺 = (^◡inl ∘ 𝐹)

**Assertion**
Ref	Expression
ctssdccl	⊢ (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆))

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.

Step	Hyp	Ref	Expression
1		ctssdccl.s	. . . 4 ⊢ 𝑆 = {𝑥 ∈ ω ∣ (𝐹‘𝑥) ∈ (inl “ 𝐴)}
2		ssrab2 3252	. . . 4 ⊢ {𝑥 ∈ ω ∣ (𝐹‘𝑥) ∈ (inl “ 𝐴)} ⊆ ω
3	1, 2	eqsstri 3199	. . 3 ⊢ 𝑆 ⊆ ω
4	3	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ ω)
5		djulf1o 7071	. . . . . . 7 ⊢ inl:V–1-1-onto→({∅} × V)
6		f1ocnv 5486	. . . . . . 7 ⊢ (inl:V–1-1-onto→({∅} × V) → ^◡inl:({∅} × V)–1-1-onto→V)
7		f1ofun 5475	. . . . . . 7 ⊢ (^◡inl:({∅} × V)–1-1-onto→V → Fun ^◡inl)
8	5, 6, 7	mp2b 8	. . . . . 6 ⊢ Fun ^◡inl
9		ctssdccl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1_o))
10		fofun 5451	. . . . . . 7 ⊢ (𝐹:ω–onto→(𝐴 ⊔ 1_o) → Fun 𝐹)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
12		funco 5268	. . . . . . 7 ⊢ ((Fun ^◡inl ∧ Fun 𝐹) → Fun (^◡inl ∘ 𝐹))
13		ctssdccl.g	. . . . . . . 8 ⊢ 𝐺 = (^◡inl ∘ 𝐹)
14	13	funeqi 5249	. . . . . . 7 ⊢ (Fun 𝐺 ↔ Fun (^◡inl ∘ 𝐹))
15	12, 14	sylibr 134	. . . . . 6 ⊢ ((Fun ^◡inl ∧ Fun 𝐹) → Fun 𝐺)
16	8, 11, 15	sylancr 414	. . . . 5 ⊢ (𝜑 → Fun 𝐺)
17		fof 5450	. . . . . . . . . . . 12 ⊢ (𝐹:ω–onto→(𝐴 ⊔ 1_o) → 𝐹:ω⟶(𝐴 ⊔ 1_o))
18	9, 17	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ω⟶(𝐴 ⊔ 1_o))
19	18	fdmd 5384	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = ω)
20	19	eleq2d 2257	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ dom 𝐹 ↔ 𝑛 ∈ ω))
21	20	anbi1d 465	. . . . . . . 8 ⊢ (𝜑 → ((𝑛 ∈ dom 𝐹 ∧ (𝐹‘𝑛) ∈ dom ^◡inl) ↔ (𝑛 ∈ ω ∧ (𝐹‘𝑛) ∈ dom ^◡inl)))
22		dmcoss 4908	. . . . . . . . . . . 12 ⊢ dom (^◡inl ∘ 𝐹) ⊆ dom 𝐹
23	22	sseli 3163	. . . . . . . . . . 11 ⊢ (𝑛 ∈ dom (^◡inl ∘ 𝐹) → 𝑛 ∈ dom 𝐹)
24	23	pm4.71ri 392	. . . . . . . . . 10 ⊢ (𝑛 ∈ dom (^◡inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹 ∧ 𝑛 ∈ dom (^◡inl ∘ 𝐹)))
25		dmfco 5597	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑛 ∈ dom 𝐹) → (𝑛 ∈ dom (^◡inl ∘ 𝐹) ↔ (𝐹‘𝑛) ∈ dom ^◡inl))
26	25	pm5.32da 452	. . . . . . . . . 10 ⊢ (Fun 𝐹 → ((𝑛 ∈ dom 𝐹 ∧ 𝑛 ∈ dom (^◡inl ∘ 𝐹)) ↔ (𝑛 ∈ dom 𝐹 ∧ (𝐹‘𝑛) ∈ dom ^◡inl)))
27	24, 26	bitrid 192	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝑛 ∈ dom (^◡inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹 ∧ (𝐹‘𝑛) ∈ dom ^◡inl)))
28	11, 27	syl 14	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ dom (^◡inl ∘ 𝐹) ↔ (𝑛 ∈ dom 𝐹 ∧ (𝐹‘𝑛) ∈ dom ^◡inl)))
29		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴)) → (𝐹‘𝑛) ∈ (inl “ 𝐴))
30		imassrn 4993	. . . . . . . . . . . . . 14 ⊢ (inl “ 𝐴) ⊆ ran inl
31	30	sseli 3163	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ (inl “ 𝐴) → (𝐹‘𝑛) ∈ ran inl)
32	31	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴)) → (𝐹‘𝑛) ∈ ran inl)
33		df-rn 4649	. . . . . . . . . . . . 13 ⊢ ran inl = dom ^◡inl
34	33	eleq2i 2254	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ ran inl ↔ (𝐹‘𝑛) ∈ dom ^◡inl)
35	32, 34	sylib 122	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴)) → (𝐹‘𝑛) ∈ dom ^◡inl)
36	29, 35	2thd 175	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴)) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ↔ (𝐹‘𝑛) ∈ dom ^◡inl))
37		djuin 7077	. . . . . . . . . . . . . 14 ⊢ ((inl “ 𝐴) ∩ (inr “ 1_o)) = ∅
38		disjel 3489	. . . . . . . . . . . . . 14 ⊢ ((((inl “ 𝐴) ∩ (inr “ 1_o)) = ∅ ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴)) → ¬ (𝐹‘𝑛) ∈ (inr “ 1_o))
39	37, 38	mpan 424	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ (inl “ 𝐴) → ¬ (𝐹‘𝑛) ∈ (inr “ 1_o))
40	39	con2i 628	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ (inr “ 1_o) → ¬ (𝐹‘𝑛) ∈ (inl “ 𝐴))
41	40	adantl 277	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inr “ 1_o)) → ¬ (𝐹‘𝑛) ∈ (inl “ 𝐴))
42		djuin 7077	. . . . . . . . . . . . . . . 16 ⊢ ((inl “ V) ∩ (inr “ 1_o)) = ∅
43		disjel 3489	. . . . . . . . . . . . . . . 16 ⊢ ((((inl “ V) ∩ (inr “ 1_o)) = ∅ ∧ (𝐹‘𝑛) ∈ (inl “ V)) → ¬ (𝐹‘𝑛) ∈ (inr “ 1_o))
44	42, 43	mpan 424	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑛) ∈ (inl “ V) → ¬ (𝐹‘𝑛) ∈ (inr “ 1_o))
45		dfrn4 5101	. . . . . . . . . . . . . . 15 ⊢ ran inl = (inl “ V)
46	44, 45	eleq2s 2282	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑛) ∈ ran inl → ¬ (𝐹‘𝑛) ∈ (inr “ 1_o))
47	46	con2i 628	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ (inr “ 1_o) → ¬ (𝐹‘𝑛) ∈ ran inl)
48	47	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inr “ 1_o)) → ¬ (𝐹‘𝑛) ∈ ran inl)
49	48, 34	sylnib 677	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inr “ 1_o)) → ¬ (𝐹‘𝑛) ∈ dom ^◡inl)
50	41, 49	2falsed 703	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inr “ 1_o)) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ↔ (𝐹‘𝑛) ∈ dom ^◡inl))
51	18	ffvelcdmda 5664	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝐹‘𝑛) ∈ (𝐴 ⊔ 1_o))
52		djuun 7080	. . . . . . . . . . . . 13 ⊢ ((inl “ 𝐴) ∪ (inr “ 1_o)) = (𝐴 ⊔ 1_o)
53	52	eleq2i 2254	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1_o)) ↔ (𝐹‘𝑛) ∈ (𝐴 ⊔ 1_o))
54	51, 53	sylibr 134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝐹‘𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1_o)))
55		elun 3288	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1_o)) ↔ ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1_o)))
56	54, 55	sylib 122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1_o)))
57	36, 50, 56	mpjaodan 799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ↔ (𝐹‘𝑛) ∈ dom ^◡inl))
58	57	pm5.32da 452	. . . . . . . 8 ⊢ (𝜑 → ((𝑛 ∈ ω ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴)) ↔ (𝑛 ∈ ω ∧ (𝐹‘𝑛) ∈ dom ^◡inl)))
59	21, 28, 58	3bitr4d 220	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ dom (^◡inl ∘ 𝐹) ↔ (𝑛 ∈ ω ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴))))
60	13	dmeqi 4840	. . . . . . . 8 ⊢ dom 𝐺 = dom (^◡inl ∘ 𝐹)
61	60	eleq2i 2254	. . . . . . 7 ⊢ (𝑛 ∈ dom 𝐺 ↔ 𝑛 ∈ dom (^◡inl ∘ 𝐹))
62		fveq2 5527	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
63	62	eleq1d 2256	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑥) ∈ (inl “ 𝐴) ↔ (𝐹‘𝑛) ∈ (inl “ 𝐴)))
64	63, 1	elrab2 2908	. . . . . . 7 ⊢ (𝑛 ∈ 𝑆 ↔ (𝑛 ∈ ω ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴)))
65	59, 61, 64	3bitr4g 223	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ dom 𝐺 ↔ 𝑛 ∈ 𝑆))
66	65	eqrdv 2185	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝑆)
67		df-fn 5231	. . . . 5 ⊢ (𝐺 Fn 𝑆 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝑆))
68	16, 66, 67	sylanbrc 417	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝑆)
69	13	fveq1i 5528	. . . . . . 7 ⊢ (𝐺‘𝑚) = ((^◡inl ∘ 𝐹)‘𝑚)
70	18	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1_o))
71		fveq2 5527	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑚 → (𝐹‘𝑥) = (𝐹‘𝑚))
72	71	eleq1d 2256	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑚 → ((𝐹‘𝑥) ∈ (inl “ 𝐴) ↔ (𝐹‘𝑚) ∈ (inl “ 𝐴)))
73	72, 1	elrab2 2908	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝑆 ↔ (𝑚 ∈ ω ∧ (𝐹‘𝑚) ∈ (inl “ 𝐴)))
74	73	biimpi 120	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝑆 → (𝑚 ∈ ω ∧ (𝐹‘𝑚) ∈ (inl “ 𝐴)))
75	74	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → (𝑚 ∈ ω ∧ (𝐹‘𝑚) ∈ (inl “ 𝐴)))
76	75	simpld 112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ ω)
77		fvco3 5600	. . . . . . . 8 ⊢ ((𝐹:ω⟶(𝐴 ⊔ 1_o) ∧ 𝑚 ∈ ω) → ((^◡inl ∘ 𝐹)‘𝑚) = (^◡inl‘(𝐹‘𝑚)))
78	70, 76, 77	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → ((^◡inl ∘ 𝐹)‘𝑚) = (^◡inl‘(𝐹‘𝑚)))
79	69, 78	eqtrid 2232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → (𝐺‘𝑚) = (^◡inl‘(𝐹‘𝑚)))
80		f1ofun 5475	. . . . . . . . . 10 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl)
81	5, 80	ax-mp 5	. . . . . . . . 9 ⊢ Fun inl
82		fvelima 5580	. . . . . . . . 9 ⊢ ((Fun inl ∧ (𝐹‘𝑚) ∈ (inl “ 𝐴)) → ∃𝑧 ∈ 𝐴 (inl‘𝑧) = (𝐹‘𝑚))
83	81, 82	mpan 424	. . . . . . . 8 ⊢ ((𝐹‘𝑚) ∈ (inl “ 𝐴) → ∃𝑧 ∈ 𝐴 (inl‘𝑧) = (𝐹‘𝑚))
84	75, 83	simpl2im 386	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → ∃𝑧 ∈ 𝐴 (inl‘𝑧) = (𝐹‘𝑚))
85		simprr 531	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inl‘𝑧) = (𝐹‘𝑚))) → (inl‘𝑧) = (𝐹‘𝑚))
86	85	fveq2d 5531	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inl‘𝑧) = (𝐹‘𝑚))) → (^◡inl‘(inl‘𝑧)) = (^◡inl‘(𝐹‘𝑚)))
87		vex 2752	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
88		f1ocnvfv1 5791	. . . . . . . . . 10 ⊢ ((inl:V–1-1-onto→({∅} × V) ∧ 𝑧 ∈ V) → (^◡inl‘(inl‘𝑧)) = 𝑧)
89	5, 87, 88	mp2an 426	. . . . . . . . 9 ⊢ (^◡inl‘(inl‘𝑧)) = 𝑧
90		simprl 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inl‘𝑧) = (𝐹‘𝑚))) → 𝑧 ∈ 𝐴)
91	89, 90	eqeltrid 2274	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inl‘𝑧) = (𝐹‘𝑚))) → (^◡inl‘(inl‘𝑧)) ∈ 𝐴)
92	86, 91	eqeltrrd 2265	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inl‘𝑧) = (𝐹‘𝑚))) → (^◡inl‘(𝐹‘𝑚)) ∈ 𝐴)
93	84, 92	rexlimddv 2609	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → (^◡inl‘(𝐹‘𝑚)) ∈ 𝐴)
94	79, 93	eqeltrd 2264	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → (𝐺‘𝑚) ∈ 𝐴)
95	94	ralrimiva 2560	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ 𝑆 (𝐺‘𝑚) ∈ 𝐴)
96		ffnfv 5687	. . . 4 ⊢ (𝐺:𝑆⟶𝐴 ↔ (𝐺 Fn 𝑆 ∧ ∀𝑚 ∈ 𝑆 (𝐺‘𝑚) ∈ 𝐴))
97	68, 95, 96	sylanbrc 417	. . 3 ⊢ (𝜑 → 𝐺:𝑆⟶𝐴)
98		djulcl 7064	. . . . . . . 8 ⊢ (𝑚 ∈ 𝐴 → (inl‘𝑚) ∈ (𝐴 ⊔ 1_o))
99		foelrn 5766	. . . . . . . . . 10 ⊢ ((𝐹:ω–onto→(𝐴 ⊔ 1_o) ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1_o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹‘𝑦))
100	9, 99	sylan 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1_o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹‘𝑦))
101		df-rex 2471	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ω (inl‘𝑚) = (𝐹‘𝑦) ↔ ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦)))
102	100, 101	sylib 122	. . . . . . . 8 ⊢ ((𝜑 ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1_o)) → ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦)))
103	98, 102	sylan2 286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦)))
104		fveq2 5527	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
105	104	eleq1d 2256	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ∈ (inl “ 𝐴) ↔ (𝐹‘𝑦) ∈ (inl “ 𝐴)))
106		simprl 529	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦))) → 𝑦 ∈ ω)
107		simprr 531	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦))) → (inl‘𝑚) = (𝐹‘𝑦))
108		vex 2752	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
109		f1odm 5477	. . . . . . . . . . . . . . . . 17 ⊢ (inl:V–1-1-onto→({∅} × V) → dom inl = V)
110	5, 109	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom inl = V
111	108, 110	eleqtrri 2263	. . . . . . . . . . . . . . 15 ⊢ 𝑚 ∈ dom inl
112		funfvima 5761	. . . . . . . . . . . . . . 15 ⊢ ((Fun inl ∧ 𝑚 ∈ dom inl) → (𝑚 ∈ 𝐴 → (inl‘𝑚) ∈ (inl “ 𝐴)))
113	81, 111, 112	mp2an 426	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝐴 → (inl‘𝑚) ∈ (inl “ 𝐴))
114	113	ad2antlr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦))) → (inl‘𝑚) ∈ (inl “ 𝐴))
115	107, 114	eqeltrrd 2265	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦))) → (𝐹‘𝑦) ∈ (inl “ 𝐴))
116	105, 106, 115	elrabd 2907	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦))) → 𝑦 ∈ {𝑥 ∈ ω ∣ (𝐹‘𝑥) ∈ (inl “ 𝐴)})
117	116, 1	eleqtrrdi 2281	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦))) → 𝑦 ∈ 𝑆)
118	117, 107	jca 306	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦))) → (𝑦 ∈ 𝑆 ∧ (inl‘𝑚) = (𝐹‘𝑦)))
119	118	ex 115	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ((𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦)) → (𝑦 ∈ 𝑆 ∧ (inl‘𝑚) = (𝐹‘𝑦))))
120	119	eximdv 1890	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦)) → ∃𝑦(𝑦 ∈ 𝑆 ∧ (inl‘𝑚) = (𝐹‘𝑦))))
121	103, 120	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ∃𝑦(𝑦 ∈ 𝑆 ∧ (inl‘𝑚) = (𝐹‘𝑦)))
122		df-rex 2471	. . . . . 6 ⊢ (∃𝑦 ∈ 𝑆 (inl‘𝑚) = (𝐹‘𝑦) ↔ ∃𝑦(𝑦 ∈ 𝑆 ∧ (inl‘𝑚) = (𝐹‘𝑦)))
123	121, 122	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ∃𝑦 ∈ 𝑆 (inl‘𝑚) = (𝐹‘𝑦))
124		f1ocnvfv1 5791	. . . . . . . . . 10 ⊢ ((inl:V–1-1-onto→({∅} × V) ∧ 𝑚 ∈ V) → (^◡inl‘(inl‘𝑚)) = 𝑚)
125	5, 108, 124	mp2an 426	. . . . . . . . 9 ⊢ (^◡inl‘(inl‘𝑚)) = 𝑚
126		simpr 110	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inl‘𝑚) = (𝐹‘𝑦)) → (inl‘𝑚) = (𝐹‘𝑦))
127	126	fveq2d 5531	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inl‘𝑚) = (𝐹‘𝑦)) → (^◡inl‘(inl‘𝑚)) = (^◡inl‘(𝐹‘𝑦)))
128	125, 127	eqtr3id 2234	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inl‘𝑚) = (𝐹‘𝑦)) → 𝑚 = (^◡inl‘(𝐹‘𝑦)))
129	13	fveq1i 5528	. . . . . . . . . 10 ⊢ (𝐺‘𝑦) = ((^◡inl ∘ 𝐹)‘𝑦)
130	18	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1_o))
131	3	sseli 3163	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ω)
132	131	adantl 277	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ω)
133		fvco3 5600	. . . . . . . . . . 11 ⊢ ((𝐹:ω⟶(𝐴 ⊔ 1_o) ∧ 𝑦 ∈ ω) → ((^◡inl ∘ 𝐹)‘𝑦) = (^◡inl‘(𝐹‘𝑦)))
134	130, 132, 133	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → ((^◡inl ∘ 𝐹)‘𝑦) = (^◡inl‘(𝐹‘𝑦)))
135	129, 134	eqtrid 2232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = (^◡inl‘(𝐹‘𝑦)))
136	135	adantr 276	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inl‘𝑚) = (𝐹‘𝑦)) → (𝐺‘𝑦) = (^◡inl‘(𝐹‘𝑦)))
137	128, 136	eqtr4d 2223	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inl‘𝑚) = (𝐹‘𝑦)) → 𝑚 = (𝐺‘𝑦))
138	137	ex 115	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → ((inl‘𝑚) = (𝐹‘𝑦) → 𝑚 = (𝐺‘𝑦)))
139	138	reximdva 2589	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (∃𝑦 ∈ 𝑆 (inl‘𝑚) = (𝐹‘𝑦) → ∃𝑦 ∈ 𝑆 𝑚 = (𝐺‘𝑦)))
140	123, 139	mpd 13	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ∃𝑦 ∈ 𝑆 𝑚 = (𝐺‘𝑦))
141	140	ralrimiva 2560	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝐴 ∃𝑦 ∈ 𝑆 𝑚 = (𝐺‘𝑦))
142		dffo3 5676	. . 3 ⊢ (𝐺:𝑆–onto→𝐴 ↔ (𝐺:𝑆⟶𝐴 ∧ ∀𝑚 ∈ 𝐴 ∃𝑦 ∈ 𝑆 𝑚 = (𝐺‘𝑦)))
143	97, 141, 142	sylanbrc 417	. 2 ⊢ (𝜑 → 𝐺:𝑆–onto→𝐴)
144	53, 55	bitr3i 186	. . . . . . 7 ⊢ ((𝐹‘𝑛) ∈ (𝐴 ⊔ 1_o) ↔ ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1_o)))
145	51, 144	sylib 122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1_o)))
146	40	orim2i 762	. . . . . 6 ⊢ (((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1_o)) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ ¬ (𝐹‘𝑛) ∈ (inl “ 𝐴)))
147	145, 146	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ ¬ (𝐹‘𝑛) ∈ (inl “ 𝐴)))
148		df-dc 836	. . . . 5 ⊢ (DECID (𝐹‘𝑛) ∈ (inl “ 𝐴) ↔ ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ ¬ (𝐹‘𝑛) ∈ (inl “ 𝐴)))
149	147, 148	sylibr 134	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → DECID (𝐹‘𝑛) ∈ (inl “ 𝐴))
150		ibar 301	. . . . . . 7 ⊢ (𝑛 ∈ ω → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ↔ (𝑛 ∈ ω ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴))))
151	150	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ↔ (𝑛 ∈ ω ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴))))
152	151, 64	bitr4di 198	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ↔ 𝑛 ∈ 𝑆))
153	152	dcbid 839	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (DECID (𝐹‘𝑛) ∈ (inl “ 𝐴) ↔ DECID 𝑛 ∈ 𝑆))
154	149, 153	mpbid 147	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → DECID 𝑛 ∈ 𝑆)
155	154	ralrimiva 2560	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
156	4, 143, 155	3jca 1178	1 ⊢ (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 ∧ w3a 979 = wceq 1363 ∃wex 1502 ∈ wcel 2158 ∀wral 2465 ∃wrex 2466 {crab 2469 Vcvv 2749 ∪ cun 3139 ∩ cin 3140 ⊆ wss 3141 ∅c0 3434 {csn 3604 ωcom 4601 × cxp 4636 ^◡ccnv 4637 dom cdm 4638 ran crn 4639 “ cima 4641 ∘ ccom 4642 Fun wfun 5222 Fn wfn 5223 ⟶wf 5224 –onto→wfo 5226 –1-1-onto→wf1o 5227 ‘cfv 5228 1_oc1o 6424 ⊔ cdju 7050 inlcinl 7058 inrcinr 7059

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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445

This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-1st 6155 df-2nd 6156 df-1o 6431 df-dju 7051 df-inl 7060 df-inr 7061

This theorem is referenced by: ctssdclemr 7125 ctiunct 12455

W3C validator