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Theorem setsn0fun 12552
Description: The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set) is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
Hypotheses
Ref Expression
setsn0fun.s  |-  ( ph  ->  S Struct  X )
setsn0fun.i  |-  ( ph  ->  I  e.  U )
setsn0fun.e  |-  ( ph  ->  E  e.  W )
Assertion
Ref Expression
setsn0fun  |-  ( ph  ->  Fun  ( ( S sSet  <. I ,  E >. ) 
\  { (/) } ) )

Proof of Theorem setsn0fun
StepHypRef Expression
1 setsn0fun.s . 2  |-  ( ph  ->  S Struct  X )
2 structn0fun 12528 . . 3  |-  ( S Struct  X  ->  Fun  ( S  \  { (/) } ) )
3 setsn0fun.i . . . . 5  |-  ( ph  ->  I  e.  U )
4 setsn0fun.e . . . . 5  |-  ( ph  ->  E  e.  W )
5 structex 12527 . . . . . . 7  |-  ( S Struct  X  ->  S  e.  _V )
6 setsfun0 12551 . . . . . . 7  |-  ( ( ( S  e.  _V  /\ 
Fun  ( S  \  { (/) } ) )  /\  ( I  e.  U  /\  E  e.  W ) )  ->  Fun  ( ( S sSet  <. I ,  E >. )  \  { (/) } ) )
75, 6sylanl1 402 . . . . . 6  |-  ( ( ( S Struct  X  /\  Fun  ( S  \  { (/)
} ) )  /\  ( I  e.  U  /\  E  e.  W
) )  ->  Fun  ( ( S sSet  <. I ,  E >. )  \  { (/) } ) )
87expcom 116 . . . . 5  |-  ( ( I  e.  U  /\  E  e.  W )  ->  ( ( S Struct  X  /\  Fun  ( S  \  { (/) } ) )  ->  Fun  ( ( S sSet  <. I ,  E >. )  \  { (/) } ) ) )
93, 4, 8syl2anc 411 . . . 4  |-  ( ph  ->  ( ( S Struct  X  /\  Fun  ( S  \  { (/) } ) )  ->  Fun  ( ( S sSet  <. I ,  E >. )  \  { (/) } ) ) )
109com12 30 . . 3  |-  ( ( S Struct  X  /\  Fun  ( S  \  { (/) } ) )  ->  ( ph  ->  Fun  ( ( S sSet  <. I ,  E >. ) 
\  { (/) } ) ) )
112, 10mpdan 421 . 2  |-  ( S Struct  X  ->  ( ph  ->  Fun  ( ( S sSet  <. I ,  E >. )  \  { (/) } ) ) )
121, 11mpcom 36 1  |-  ( ph  ->  Fun  ( ( S sSet  <. I ,  E >. ) 
\  { (/) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   _Vcvv 2752    \ cdif 3141   (/)c0 3437   {csn 3607   <.cop 3610   class class class wbr 4018   Fun wfun 5229  (class class class)co 5897   Struct cstr 12511   sSet csts 12513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-res 4656  df-iota 5196  df-fun 5237  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-struct 12517  df-sets 12522
This theorem is referenced by: (None)
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