Metamath Proof Explorer


Theorem caragenunicl

Description: The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of Fremlin1 p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses caragenunicl.o φ O OutMeas
caragenunicl.s S = CaraGen O
caragenunicl.y φ X S
caragenunicl.ctb φ X ω
Assertion caragenunicl φ X S

Proof

Step Hyp Ref Expression
1 caragenunicl.o φ O OutMeas
2 caragenunicl.s S = CaraGen O
3 caragenunicl.y φ X S
4 caragenunicl.ctb φ X ω
5 unieq X = X =
6 uni0 =
7 5 6 eqtrdi X = X =
8 7 adantl φ X = X =
9 1 2 caragen0 φ S
10 9 adantr φ X = S
11 8 10 eqeltrd φ X = X S
12 simpl φ ¬ X = φ
13 neqne ¬ X = X
14 13 adantl φ ¬ X = X
15 simpr φ X X
16 reldom Rel
17 brrelex1 Rel X ω X V
18 16 17 mpan X ω X V
19 4 18 syl φ X V
20 19 adantr φ X X V
21 0sdomg X V X X
22 20 21 syl φ X X X
23 15 22 mpbird φ X X
24 nnenom ω
25 24 ensymi ω
26 25 a1i φ ω
27 domentr X ω ω X
28 4 26 27 syl2anc φ X
29 28 adantr φ X X
30 fodomr X X f f : onto X
31 23 29 30 syl2anc φ X f f : onto X
32 founiiun f : onto X X = n f n
33 32 adantl φ f : onto X X = n f n
34 1 adantr φ f : onto X O OutMeas
35 1zzd φ f : onto X 1
36 nnuz = 1
37 fof f : onto X f : X
38 37 adantl φ f : onto X f : X
39 3 adantr φ f : onto X X S
40 38 39 fssd φ f : onto X f : S
41 34 2 35 36 40 carageniuncl φ f : onto X n f n S
42 33 41 eqeltrd φ f : onto X X S
43 42 ex φ f : onto X X S
44 43 adantr φ X f : onto X X S
45 44 exlimdv φ X f f : onto X X S
46 31 45 mpd φ X X S
47 12 14 46 syl2anc φ ¬ X = X S
48 11 47 pm2.61dan φ X S