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Theorem sbcfung 5624
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 3342 . . 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 4940 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  F  <->  Rel  [_ A  /  x ]_ F ) )
3 sbcal 3349 . . . . 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 3350 . . . . . . 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 3349 . . . . . . . . 9  |-  ( [. A  /  x ]. A. z ( w F z  ->  z  =  y )  <->  A. z [. A  /  x ]. ( w F z  ->  z  =  y ) )
6 sbcimg 3341 . . . . . . . . . . 11  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( w F z  ->  z  =  y )  <->  ( [. A  /  x ]. w F z  ->  [. A  /  x ]. z  =  y ) ) )
7 sbcbr123 4475 . . . . . . . . . . . . 13  |-  ( [. A  /  x ]. w F z  <->  [_ A  /  x ]_ w [_ A  /  x ]_ F [_ A  /  x ]_ z
)
8 csbconstg 3408 . . . . . . . . . . . . . 14  |-  ( A  e.  V  ->  [_ A  /  x ]_ w  =  w )
9 csbconstg 3408 . . . . . . . . . . . . . 14  |-  ( A  e.  V  ->  [_ A  /  x ]_ z  =  z )
108, 9breq12d 4436 . . . . . . . . . . . . 13  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ w [_ A  /  x ]_ F [_ A  /  x ]_ z  <->  w [_ A  /  x ]_ F
z ) )
117, 10syl5bb 260 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( [. A  /  x ]. w F z  <->  w [_ A  /  x ]_ F
z ) )
12 sbcg 3365 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  =  y  <->  z  =  y ) )
1311, 12imbi12d 321 . . . . . . . . . . 11  |-  ( A  e.  V  ->  (
( [. A  /  x ]. w F z  ->  [. A  /  x ]. z  =  y
)  <->  ( w [_ A  /  x ]_ F
z  ->  z  =  y ) ) )
146, 13bitrd 256 . . . . . . . . . 10  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( w F z  ->  z  =  y )  <->  ( w [_ A  /  x ]_ F
z  ->  z  =  y ) ) )
1514albidv 1761 . . . . . . . . 9  |-  ( A  e.  V  ->  ( A. z [. A  /  x ]. ( w F z  ->  z  =  y )  <->  A. z
( w [_ A  /  x ]_ F z  ->  z  =  y ) ) )
165, 15syl5bb 260 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. A. z ( w F z  ->  z  =  y )  <->  A. z
( w [_ A  /  x ]_ F z  ->  z  =  y ) ) )
1716exbidv 1762 . . . . . . 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 260 . . . . . 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 1761 . . . . 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 260 . . . 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 715 . . 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 260 . 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 5612 . . 3  |-  ( Fun 
F  <->  ( Rel  F  /\  A. w E. y A. z ( w F z  ->  z  =  y ) ) )
2423sbcbii 3355 . 2  |-  ( [. A  /  x ]. Fun  F  <->  [. A  /  x ]. ( Rel  F  /\  A. w E. y A. z ( w F z  ->  z  =  y ) ) )
25 dffun3 5612 . 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 291 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. Fun  F  <->  Fun  [_ A  /  x ]_ F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1657    e. wcel 1872   [.wsbc 3299   [_csb 3395   class class class wbr 4423   Rel wrel 4858   Fun wfun 5595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-id 4768  df-rel 4860  df-cnv 4861  df-co 4862  df-fun 5603
This theorem is referenced by:  sbcfng  5743  esum2dlem  28921
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