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Theorem mbfmptcl 22022
Description: Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.)
Hypotheses
Ref Expression
mbfmptcl.1  |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )
mbfmptcl.2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
Assertion
Ref Expression
mbfmptcl  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
Distinct variable groups:    x, A    ph, x
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem mbfmptcl
StepHypRef Expression
1 mbfmptcl.1 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )
2 mbff 22012 . . . . 5  |-  ( ( x  e.  A  |->  B )  e. MblFn  ->  ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> CC )
31, 2syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> CC )
4 mbfmptcl.2 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
54ralrimiva 2857 . . . . . 6  |-  ( ph  ->  A. x  e.  A  B  e.  V )
6 dmmptg 5494 . . . . . 6  |-  ( A. x  e.  A  B  e.  V  ->  dom  (
x  e.  A  |->  B )  =  A )
75, 6syl 16 . . . . 5  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
87feq2d 5708 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> CC  <->  ( x  e.  A  |->  B ) : A --> CC ) )
93, 8mpbid 210 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> CC )
10 eqid 2443 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
1110fmpt 6037 . . 3  |-  ( A. x  e.  A  B  e.  CC  <->  ( x  e.  A  |->  B ) : A --> CC )
129, 11sylibr 212 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  CC )
1312r19.21bi 2812 1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793    |-> cmpt 4495   dom cdm 4989   -->wf 5574   CCcc 9493  MblFncmbf 22001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-pm 7425  df-mbf 22006
This theorem is referenced by:  mbfss  22031  mbfneg  22035  mbfmulc2  22048  mbflim  22053  itgcnlem  22174  itgcnval  22184  itgre  22185  itgim  22186  iblneg  22187  itgneg  22188  iblss  22189  iblss2  22190  ibladd  22205  iblsub  22206  itgadd  22209  itgsub  22210  itgfsum  22211  iblabs  22213  iblabsr  22214  iblmulc2  22215  itgmulc2  22218  itgabs  22219  itgsplit  22220  bddmulibl  22223  itgcn  22227  ditgswap  22241  ditgsplitlem  22242  ftc1a  22416  ibladdnc  30048  itgaddnc  30051  iblsubnc  30052  itgsubnc  30053  iblabsnc  30055  iblmulc2nc  30056  itgmulc2nc  30059  itgabsnc  30060
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