Metamath Proof Explorer


Theorem mbfmcst

Description: A constant function is measurable. Cf. mbfconst . (Contributed by Thierry Arnoux, 26-Jan-2017)

Ref Expression
Hypotheses mbfmcst.1 φSransigAlgebra
mbfmcst.2 φTransigAlgebra
mbfmcst.3 φF=xSA
mbfmcst.4 φAT
Assertion mbfmcst φFSMblFnμT

Proof

Step Hyp Ref Expression
1 mbfmcst.1 φSransigAlgebra
2 mbfmcst.2 φTransigAlgebra
3 mbfmcst.3 φF=xSA
4 mbfmcst.4 φAT
5 4 adantr φxSAT
6 3 5 fmpt3d φF:ST
7 unielsiga TransigAlgebraTT
8 2 7 syl φTT
9 unielsiga SransigAlgebraSS
10 1 9 syl φSS
11 8 10 elmapd φFTSF:ST
12 6 11 mpbird φFTS
13 fconstmpt S×A=xSA
14 13 cnveqi S×A-1=xSA-1
15 cnvxp S×A-1=A×S
16 14 15 eqtr3i xSA-1=A×S
17 16 imaeq1i xSA-1y=A×Sy
18 df-ima A×Sy=ranA×Sy
19 df-rn ranA×Sy=domA×Sy-1
20 17 18 19 3eqtri xSA-1y=domA×Sy-1
21 df-res A×Sy=A×Sy×V
22 inxp A×Sy×V=Ay×SV
23 inv1 SV=S
24 23 xpeq2i Ay×SV=Ay×S
25 21 22 24 3eqtri A×Sy=Ay×S
26 25 cnveqi A×Sy-1=Ay×S-1
27 26 dmeqi domA×Sy-1=domAy×S-1
28 cnvxp Ay×S-1=S×Ay
29 28 dmeqi domAy×S-1=domS×Ay
30 20 27 29 3eqtri xSA-1y=domS×Ay
31 xpeq2 Ay=S×Ay=S×
32 xp0 S×=
33 31 32 eqtrdi Ay=S×Ay=
34 33 dmeqd Ay=domS×Ay=dom
35 dm0 dom=
36 34 35 eqtrdi Ay=domS×Ay=
37 36 adantl φAy=domS×Ay=
38 0elsiga SransigAlgebraS
39 1 38 syl φS
40 39 adantr φAy=S
41 37 40 eqeltrd φAy=domS×AyS
42 30 41 eqeltrid φAy=xSA-1yS
43 dmxp AydomS×Ay=S
44 43 adantl φAydomS×Ay=S
45 10 adantr φAySS
46 44 45 eqeltrd φAydomS×AyS
47 30 46 eqeltrid φAyxSA-1yS
48 42 47 pm2.61dane φxSA-1yS
49 48 ralrimivw φyTxSA-1yS
50 3 cnveqd φF-1=xSA-1
51 50 imaeq1d φF-1y=xSA-1y
52 51 eleq1d φF-1ySxSA-1yS
53 52 ralbidv φyTF-1ySyTxSA-1yS
54 49 53 mpbird φyTF-1yS
55 1 2 ismbfm φFSMblFnμTFTSyTF-1yS
56 12 54 55 mpbir2and φFSMblFnμT