Theorem ctssdccl 7177

Description: A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7179 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 ⊔ 1o))
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 3268	. . . 4 ⊢ {𝑥 ∈ ω ∣ (𝐹‘𝑥) ∈ (inl “ 𝐴)} ⊆ ω
3	1, 2	eqsstri 3215	. . 3 ⊢ 𝑆 ⊆ ω
4	3	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ ω)
5		djulf1o 7124	. . . . . . 7 ⊢ inl:V–1-1-onto→({∅} × V)
6		f1ocnv 5517	. . . . . . 7 ⊢ (inl:V–1-1-onto→({∅} × V) → ◡inl:({∅} × V)–1-1-onto→V)
7		f1ofun 5506	. . . . . . 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 ⊔ 1o))
10		fofun 5481	. . . . . . 7 ⊢ (𝐹:ω–onto→(𝐴 ⊔ 1o) → Fun 𝐹)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
12		funco 5298	. . . . . . 7 ⊢ ((Fun ◡inl ∧ Fun 𝐹) → Fun (◡inl ∘ 𝐹))
13		ctssdccl.g	. . . . . . . 8 ⊢ 𝐺 = (◡inl ∘ 𝐹)
14	13	funeqi 5279	. . . . . . 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 5480	. . . . . . . . . . . 12 ⊢ (𝐹:ω–onto→(𝐴 ⊔ 1o) → 𝐹:ω⟶(𝐴 ⊔ 1o))
18	9, 17	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ω⟶(𝐴 ⊔ 1o))
19	18	fdmd 5414	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = ω)
20	19	eleq2d 2266	. . . . . . . . 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 4935	. . . . . . . . . . . 12 ⊢ dom (◡inl ∘ 𝐹) ⊆ dom 𝐹
23	22	sseli 3179	. . . . . . . . . . 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 5629	. . . . . . . . . . 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 5020	. . . . . . . . . . . . . 14 ⊢ (inl “ 𝐴) ⊆ ran inl
31	30	sseli 3179	. . . . . . . . . . . . 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 4674	. . . . . . . . . . . . 13 ⊢ ran inl = dom ◡inl
34	33	eleq2i 2263	. . . . . . . . . . . 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 7130	. . . . . . . . . . . . . 14 ⊢ ((inl “ 𝐴) ∩ (inr “ 1o)) = ∅
38		disjel 3505	. . . . . . . . . . . . . 14 ⊢ ((((inl “ 𝐴) ∩ (inr “ 1o)) = ∅ ∧ (𝐹‘𝑛) ∈ (inl “ 𝐴)) → ¬ (𝐹‘𝑛) ∈ (inr “ 1o))
39	37, 38	mpan 424	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ (inl “ 𝐴) → ¬ (𝐹‘𝑛) ∈ (inr “ 1o))
40	39	con2i 628	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ (inr “ 1o) → ¬ (𝐹‘𝑛) ∈ (inl “ 𝐴))
41	40	adantl 277	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inr “ 1o)) → ¬ (𝐹‘𝑛) ∈ (inl “ 𝐴))
42		djuin 7130	. . . . . . . . . . . . . . . 16 ⊢ ((inl “ V) ∩ (inr “ 1o)) = ∅
43		disjel 3505	. . . . . . . . . . . . . . . 16 ⊢ ((((inl “ V) ∩ (inr “ 1o)) = ∅ ∧ (𝐹‘𝑛) ∈ (inl “ V)) → ¬ (𝐹‘𝑛) ∈ (inr “ 1o))
44	42, 43	mpan 424	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑛) ∈ (inl “ V) → ¬ (𝐹‘𝑛) ∈ (inr “ 1o))
45		dfrn4 5130	. . . . . . . . . . . . . . 15 ⊢ ran inl = (inl “ V)
46	44, 45	eleq2s 2291	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑛) ∈ ran inl → ¬ (𝐹‘𝑛) ∈ (inr “ 1o))
47	46	con2i 628	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ (inr “ 1o) → ¬ (𝐹‘𝑛) ∈ ran inl)
48	47	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inr “ 1o)) → ¬ (𝐹‘𝑛) ∈ ran inl)
49	48, 34	sylnib 677	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inr “ 1o)) → ¬ (𝐹‘𝑛) ∈ dom ◡inl)
50	41, 49	2falsed 703	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘𝑛) ∈ (inr “ 1o)) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ↔ (𝐹‘𝑛) ∈ dom ◡inl))
51	18	ffvelcdmda 5697	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝐹‘𝑛) ∈ (𝐴 ⊔ 1o))
52		djuun 7133	. . . . . . . . . . . . 13 ⊢ ((inl “ 𝐴) ∪ (inr “ 1o)) = (𝐴 ⊔ 1o)
53	52	eleq2i 2263	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)) ↔ (𝐹‘𝑛) ∈ (𝐴 ⊔ 1o))
54	51, 53	sylibr 134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝐹‘𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)))
55		elun 3304	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∈ ((inl “ 𝐴) ∪ (inr “ 1o)) ↔ ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1o)))
56	54, 55	sylib 122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1o)))
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 4867	. . . . . . . 8 ⊢ dom 𝐺 = dom (◡inl ∘ 𝐹)
61	60	eleq2i 2263	. . . . . . 7 ⊢ (𝑛 ∈ dom 𝐺 ↔ 𝑛 ∈ dom (◡inl ∘ 𝐹))
62		fveq2 5558	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
63	62	eleq1d 2265	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑥) ∈ (inl “ 𝐴) ↔ (𝐹‘𝑛) ∈ (inl “ 𝐴)))
64	63, 1	elrab2 2923	. . . . . . 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 2194	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝑆)
67		df-fn 5261	. . . . 5 ⊢ (𝐺 Fn 𝑆 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝑆))
68	16, 66, 67	sylanbrc 417	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝑆)
69	13	fveq1i 5559	. . . . . . 7 ⊢ (𝐺‘𝑚) = ((◡inl ∘ 𝐹)‘𝑚)
70	18	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1o))
71		fveq2 5558	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑚 → (𝐹‘𝑥) = (𝐹‘𝑚))
72	71	eleq1d 2265	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑚 → ((𝐹‘𝑥) ∈ (inl “ 𝐴) ↔ (𝐹‘𝑚) ∈ (inl “ 𝐴)))
73	72, 1	elrab2 2923	. . . . . . . . . . 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 5632	. . . . . . . 8 ⊢ ((𝐹:ω⟶(𝐴 ⊔ 1o) ∧ 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 2241	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → (𝐺‘𝑚) = (◡inl‘(𝐹‘𝑚)))
80		f1ofun 5506	. . . . . . . . . 10 ⊢ (inl:V–1-1-onto→({∅} × V) → Fun inl)
81	5, 80	ax-mp 5	. . . . . . . . 9 ⊢ Fun inl
82		fvelima 5612	. . . . . . . . 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 5562	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inl‘𝑧) = (𝐹‘𝑚))) → (◡inl‘(inl‘𝑧)) = (◡inl‘(𝐹‘𝑚)))
87		vex 2766	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
88		f1ocnvfv1 5824	. . . . . . . . . 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 2283	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inl‘𝑧) = (𝐹‘𝑚))) → (◡inl‘(inl‘𝑧)) ∈ 𝐴)
92	86, 91	eqeltrrd 2274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑆) ∧ (𝑧 ∈ 𝐴 ∧ (inl‘𝑧) = (𝐹‘𝑚))) → (◡inl‘(𝐹‘𝑚)) ∈ 𝐴)
93	84, 92	rexlimddv 2619	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → (◡inl‘(𝐹‘𝑚)) ∈ 𝐴)
94	79, 93	eqeltrd 2273	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑆) → (𝐺‘𝑚) ∈ 𝐴)
95	94	ralrimiva 2570	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ 𝑆 (𝐺‘𝑚) ∈ 𝐴)
96		ffnfv 5720	. . . 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 7117	. . . . . . . 8 ⊢ (𝑚 ∈ 𝐴 → (inl‘𝑚) ∈ (𝐴 ⊔ 1o))
99		foelrn 5799	. . . . . . . . . 10 ⊢ ((𝐹:ω–onto→(𝐴 ⊔ 1o) ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹‘𝑦))
100	9, 99	sylan 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω (inl‘𝑚) = (𝐹‘𝑦))
101		df-rex 2481	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ω (inl‘𝑚) = (𝐹‘𝑦) ↔ ∃𝑦(𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦)))
102	100, 101	sylib 122	. . . . . . . 8 ⊢ ((𝜑 ∧ (inl‘𝑚) ∈ (𝐴 ⊔ 1o)) → ∃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 5558	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
105	104	eleq1d 2265	. . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
109		f1odm 5508	. . . . . . . . . . . . . . . . 17 ⊢ (inl:V–1-1-onto→({∅} × V) → dom inl = V)
110	5, 109	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom inl = V
111	108, 110	eleqtrri 2272	. . . . . . . . . . . . . . 15 ⊢ 𝑚 ∈ dom inl
112		funfvima 5794	. . . . . . . . . . . . . . 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 2274	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦))) → (𝐹‘𝑦) ∈ (inl “ 𝐴))
116	105, 106, 115	elrabd 2922	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ (𝑦 ∈ ω ∧ (inl‘𝑚) = (𝐹‘𝑦))) → 𝑦 ∈ {𝑥 ∈ ω ∣ (𝐹‘𝑥) ∈ (inl “ 𝐴)})
117	116, 1	eleqtrrdi 2290	. . . . . . . . . 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 1894	. . . . . . 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 2481	. . . . . 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 5824	. . . . . . . . . 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 5562	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inl‘𝑚) = (𝐹‘𝑦)) → (◡inl‘(inl‘𝑚)) = (◡inl‘(𝐹‘𝑦)))
128	125, 127	eqtr3id 2243	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (inl‘𝑚) = (𝐹‘𝑦)) → 𝑚 = (◡inl‘(𝐹‘𝑦)))
129	13	fveq1i 5559	. . . . . . . . . 10 ⊢ (𝐺‘𝑦) = ((◡inl ∘ 𝐹)‘𝑦)
130	18	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → 𝐹:ω⟶(𝐴 ⊔ 1o))
131	3	sseli 3179	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ω)
132	131	adantl 277	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ω)
133		fvco3 5632	. . . . . . . . . . 11 ⊢ ((𝐹:ω⟶(𝐴 ⊔ 1o) ∧ 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 2241	. . . . . . . . 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 2232	. . . . . . 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 2599	. . . . 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 2570	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝐴 ∃𝑦 ∈ 𝑆 𝑚 = (𝐺‘𝑦))
142		dffo3 5709	. . 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 ⊔ 1o) ↔ ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1o)))
145	51, 144	sylib 122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1o)))
146	40	orim2i 762	. . . . . 6 ⊢ (((𝐹‘𝑛) ∈ (inl “ 𝐴) ∨ (𝐹‘𝑛) ∈ (inr “ 1o)) → ((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 2570	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
156	4, 143, 155	3jca 1179	1 ⊢ (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479  Vcvv 2763   ∪ cun 3155   ∩ cin 3156   ⊆ wss 3157  ∅c0 3450  {csn 3622  ωcom 4626   × cxp 4661  ^◡ccnv 4662  dom cdm 4663  ran crn 4664   “ cima 4666   ∘ ccom 4667  Fun wfun 5252   Fn wfn 5253  ⟶wf 5254  –onto→wfo 5256  –1-1-onto→wf1o 5257  ‘cfv 5258  1_oc1o 6467   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114

This theorem is referenced by:  ctssdclemr  7178  ctiunct  12657