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Theorem setsn0fun 12482
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 12458 . . 3  |-  ( S Struct  X  ->  Fun  ( S  \  { (/) } ) )
3 setsn0fun.i . . . . 5  |-  ( ph  ->  I  e.  U )
4 setsn0fun.e . . . . 5  |-  ( ph  ->  E  e.  W )
5 structex 12457 . . . . . . 7  |-  ( S Struct  X  ->  S  e.  _V )
6 setsfun0 12481 . . . . . . 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 2148   _Vcvv 2737    \ cdif 3126   (/)c0 3422   {csn 3591   <.cop 3594   class class class wbr 4000   Fun wfun 5206  (class class class)co 5869   Struct cstr 12441   sSet csts 12443
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-res 4635  df-iota 5174  df-fun 5214  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-struct 12447  df-sets 12452
This theorem is referenced by: (None)
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