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Theorem fmpttd 5529
Description: Version of fmptd 5528 with inlined definition. Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by BJ, 16-Aug-2022.)
Hypothesis
Ref Expression
fmpttd.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  C )
Assertion
Ref Expression
fmpttd  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )
Distinct variable groups:    x, A    x, C    ph, x
Allowed substitution hint:    B( x)

Proof of Theorem fmpttd
StepHypRef Expression
1 fmpttd.1 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  C )
2 eqid 2115 . 2  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
31, 2fmptd 5528 1  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1463    |-> cmpt 3949   -->wf 5077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089
This theorem is referenced by:  fmpt3d  5530  ctmlemr  6945  ctssdclemn0  6947  ctssdc  6950  ismkvnex  6979  fsumf1o  11051  isumss  11052  fisumss  11053  fsumcl2lem  11059  fsumadd  11067  isumclim3  11084  isummulc2  11087  fsummulc2  11109  isumshft  11151  tgrest  12181  resttopon  12183  rest0  12191  cnpfval  12207  txcnp  12282  uptx  12285  cnmpt11  12294  bdxmet  12490  cncfmptc  12568  cncfmptid  12569  cdivcncfap  12573  mulcncf  12577  limcmpted  12588  dvfgg  12612  dvcnp2cntop  12618  dvmulxxbr  12621  subctctexmid  12888  nninffeq  12908
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