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Theorem sbcfung 5617
Description: Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcfung  |-  ( A  e.  V  ->  ( [. A  /  x ]. Fun  F  <->  Fun  [_ A  /  x ]_ F ) )

Proof of Theorem sbcfung
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcan 3379 . . 3  |-  ( [. A  /  x ]. ( Rel  F  /\  A. w E. y A. z ( w F z  -> 
z  =  y ) )  <->  ( [. A  /  x ]. Rel  F  /\  [. A  /  x ]. A. w E. y A. z ( w F z  ->  z  =  y ) ) )
2 sbcrel 5095 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  F  <->  Rel  [_ A  /  x ]_ F ) )
3 sbcal 3388 . . . . 5  |-  ( [. A  /  x ]. A. w E. y A. z
( w F z  ->  z  =  y )  <->  A. w [. A  /  x ]. E. y A. z ( w F z  ->  z  =  y ) )
4 sbcex2 3390 . . . . . . 7  |-  ( [. A  /  x ]. E. y A. z ( w F z  ->  z  =  y )  <->  E. y [. A  /  x ]. A. z ( w F z  ->  z  =  y ) )
5 sbcal 3388 . . . . . . . . 9  |-  ( [. A  /  x ]. A. z ( w F z  ->  z  =  y )  <->  A. z [. A  /  x ]. ( w F z  ->  z  =  y ) )
6 sbcimg 3378 . . . . . . . . . . 11  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( w F z  ->  z  =  y )  <->  ( [. A  /  x ]. w F z  ->  [. A  /  x ]. z  =  y ) ) )
7 sbcbr123 4504 . . . . . . . . . . . . 13  |-  ( [. A  /  x ]. w F z  <->  [_ A  /  x ]_ w [_ A  /  x ]_ F [_ A  /  x ]_ z
)
8 csbconstg 3453 . . . . . . . . . . . . . 14  |-  ( A  e.  V  ->  [_ A  /  x ]_ w  =  w )
9 csbconstg 3453 . . . . . . . . . . . . . 14  |-  ( A  e.  V  ->  [_ A  /  x ]_ z  =  z )
108, 9breq12d 4466 . . . . . . . . . . . . 13  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ w [_ A  /  x ]_ F [_ A  /  x ]_ z  <->  w [_ A  /  x ]_ F
z ) )
117, 10syl5bb 257 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( [. A  /  x ]. w F z  <->  w [_ A  /  x ]_ F
z ) )
12 sbcg 3410 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  =  y  <->  z  =  y ) )
1311, 12imbi12d 320 . . . . . . . . . . 11  |-  ( A  e.  V  ->  (
( [. A  /  x ]. w F z  ->  [. A  /  x ]. z  =  y
)  <->  ( w [_ A  /  x ]_ F
z  ->  z  =  y ) ) )
146, 13bitrd 253 . . . . . . . . . 10  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( w F z  ->  z  =  y )  <->  ( w [_ A  /  x ]_ F
z  ->  z  =  y ) ) )
1514albidv 1689 . . . . . . . . 9  |-  ( A  e.  V  ->  ( A. z [. A  /  x ]. ( w F z  ->  z  =  y )  <->  A. z
( w [_ A  /  x ]_ F z  ->  z  =  y ) ) )
165, 15syl5bb 257 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. A. z ( w F z  ->  z  =  y )  <->  A. z
( w [_ A  /  x ]_ F z  ->  z  =  y ) ) )
1716exbidv 1690 . . . . . . 7  |-  ( A  e.  V  ->  ( E. y [. A  /  x ]. A. z ( w F z  -> 
z  =  y )  <->  E. y A. z ( w [_ A  /  x ]_ F z  -> 
z  =  y ) ) )
184, 17syl5bb 257 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. y A. z
( w F z  ->  z  =  y )  <->  E. y A. z
( w [_ A  /  x ]_ F z  ->  z  =  y ) ) )
1918albidv 1689 . . . . 5  |-  ( A  e.  V  ->  ( A. w [. A  /  x ]. E. y A. z ( w F z  ->  z  =  y )  <->  A. w E. y A. z ( w [_ A  /  x ]_ F z  -> 
z  =  y ) ) )
203, 19syl5bb 257 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. A. w E. y A. z ( w F z  ->  z  =  y )  <->  A. w E. y A. z ( w [_ A  /  x ]_ F z  -> 
z  =  y ) ) )
212, 20anbi12d 710 . . 3  |-  ( A  e.  V  ->  (
( [. A  /  x ]. Rel  F  /\  [. A  /  x ]. A. w E. y A. z ( w F z  -> 
z  =  y ) )  <->  ( Rel  [_ A  /  x ]_ F  /\  A. w E. y A. z ( w [_ A  /  x ]_ F
z  ->  z  =  y ) ) ) )
221, 21syl5bb 257 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( Rel  F  /\  A. w E. y A. z ( w F z  ->  z  =  y ) )  <->  ( Rel  [_ A  /  x ]_ F  /\  A. w E. y A. z ( w
[_ A  /  x ]_ F z  ->  z  =  y ) ) ) )
23 dffun3 5605 . . 3  |-  ( Fun 
F  <->  ( Rel  F  /\  A. w E. y A. z ( w F z  ->  z  =  y ) ) )
2423sbcbii 3396 . 2  |-  ( [. A  /  x ]. Fun  F  <->  [. A  /  x ]. ( Rel  F  /\  A. w E. y A. z ( w F z  ->  z  =  y ) ) )
25 dffun3 5605 . 2  |-  ( Fun  [_ A  /  x ]_ F  <->  ( Rel  [_ A  /  x ]_ F  /\  A. w E. y A. z ( w [_ A  /  x ]_ F
z  ->  z  =  y ) ) )
2622, 24, 253bitr4g 288 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. Fun  F  <->  Fun  [_ A  /  x ]_ F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377   E.wex 1596    e. wcel 1767   [.wsbc 3336   [_csb 3440   class class class wbr 4453   Rel wrel 5010   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-id 4801  df-rel 5012  df-cnv 5013  df-co 5014  df-fun 5596
This theorem is referenced by:  sbcfng  5734
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