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Theorem mptexd 5723
Description: If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 5721. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypothesis
Ref Expression
mptexd.1  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
mptexd  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  _V )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)    V( x)

Proof of Theorem mptexd
StepHypRef Expression
1 mptexd.1 . 2  |-  ( ph  ->  A  e.  V )
2 mptexg 5721 . 2  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
31, 2syl 14 1  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141   _Vcvv 2730    |-> cmpt 4050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206
This theorem is referenced by:  grpinvfvalg  12745  grpsubval  12749
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