Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isibl Structured version   Unicode version

Theorem isibl 19657
 Description: The predicate " is integrable". The "integrable" predicate corresponds roughly to the range of validity of , which is to say that the expression doesn't make sense unless . (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
isibl.1
isibl.2
isibl.3
isibl.4
Assertion
Ref Expression
isibl MblFn
Distinct variable groups:   ,,   ,   ,,   ,,
Allowed substitution hints:   ()   (,)   (,)

Proof of Theorem isibl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5742 . . . . . . . . 9
2 nfcv 2572 . . . . . . . . 9
3 breq2 4216 . . . . . . . . . . 11
43anbi2d 685 . . . . . . . . . 10
5 id 20 . . . . . . . . . 10
6 eqidd 2437 . . . . . . . . . 10
74, 5, 6ifbieq12d 3761 . . . . . . . . 9
81, 2, 7csbief 3292 . . . . . . . 8
9 dmeq 5070 . . . . . . . . . . 11
109eleq2d 2503 . . . . . . . . . 10
11 fveq1 5727 . . . . . . . . . . . . 13
1211oveq1d 6096 . . . . . . . . . . . 12
1312fveq2d 5732 . . . . . . . . . . 11
1413breq2d 4224 . . . . . . . . . 10
1510, 14anbi12d 692 . . . . . . . . 9
16 eqidd 2437 . . . . . . . . 9
1715, 13, 16ifbieq12d 3761 . . . . . . . 8
188, 17syl5eq 2480 . . . . . . 7
1918mpteq2dv 4296 . . . . . 6
2019fveq2d 5732 . . . . 5
2120eleq1d 2502 . . . 4
2221ralbidv 2725 . . 3
23 df-ibl 19515 . . 3 MblFn
2422, 23elrab2 3094 . 2 MblFn
25 isibl.3 . . . . . . . . . . . 12
2625eleq2d 2503 . . . . . . . . . . 11
2726anbi1d 686 . . . . . . . . . 10
2827ifbid 3757 . . . . . . . . 9
29 isibl.4 . . . . . . . . . . . . 13
3029oveq1d 6096 . . . . . . . . . . . 12
3130fveq2d 5732 . . . . . . . . . . 11
32 isibl.2 . . . . . . . . . . 11
3331, 32eqtr4d 2471 . . . . . . . . . 10
3433ibllem 19656 . . . . . . . . 9
3528, 34eqtrd 2468 . . . . . . . 8
3635mpteq2dv 4296 . . . . . . 7
37 isibl.1 . . . . . . 7
3836, 37eqtr4d 2471 . . . . . 6
3938fveq2d 5732 . . . . 5
4039eleq1d 2502 . . . 4
4140ralbidv 2725 . . 3
4241anbi2d 685 . 2 MblFn MblFn
4324, 42syl5bb 249 1 MblFn
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2705  csb 3251  cif 3739   class class class wbr 4212   cmpt 4266   cdm 4878  cfv 5454  (class class class)co 6081  cr 8989  cc0 8990  ci 8992   cle 9121   cdiv 9677  c3 10050  cfz 11043  cexp 11382  cre 11902  MblFncmbf 19506  citg2 19508  cibl 19509 This theorem is referenced by:  isibl2  19658  ibl0  19678 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-dm 4888  df-iota 5418  df-fv 5462  df-ov 6084  df-ibl 19515
 Copyright terms: Public domain W3C validator