Theorem sbcfng 5159

Description: Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Assertion

Ref	Expression
sbcfng	|- ( X e. V -> ( [. X / x ]. F Fn A <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )

Distinct variable groups:   x, V   x, X

Allowed substitution hints:   A( x)   F( x)

Proof of Theorem sbcfng

Step	Hyp	Ref	Expression
1		df-fn 5018	. . . 4 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2	1	a1i 9	. . 3 |- ( X e. V -> ( F Fn A <-> ( Fun F /\ dom F = A ) ) )
3	2	sbcbidv 2897	. 2 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [. X / x ]. ( Fun F /\ dom F = A ) ) )
4		sbcfung 5039	. . . 4 |- ( X e. V -> ( [. X / x ]. Fun F <-> Fun [_ X / x ]_ F ) )
5		sbceqg 2947	. . . . 5 |- ( X e. V -> ( [. X / x ]. dom F = A <-> [_ X / x ]_ dom F = [_ X / x ]_ A ) )
6		csbdmg 4630	. . . . . 6 |- ( X e. V -> [_ X / x ]_ dom F = dom [_ X / x ]_ F )
7	6	eqeq1d 2096	. . . . 5 |- ( X e. V -> ( [_ X / x ]_ dom F = [_ X / x ]_ A <-> dom [_ X / x ]_ F = [_ X / x ]_ A ) )
8	5, 7	bitrd 186	. . . 4 |- ( X e. V -> ( [. X / x ]. dom F = A <-> dom [_ X / x ]_ F = [_ X / x ]_ A ) )
9	4, 8	anbi12d 457	. . 3 |- ( X e. V -> ( ( [. X / x ]. Fun F /\ [. X / x ]. dom F = A ) <-> ( Fun [_ X / x ]_ F /\ dom [_ X / x ]_ F = [_ X / x ]_ A ) ) )
10		sbcan 2881	. . 3 |- ( [. X / x ]. ( Fun F /\ dom F = A ) <-> ( [. X / x ]. Fun F /\ [. X / x ]. dom F = A ) )
11		df-fn 5018	. . 3 |- ( [_ X / x ]_ F Fn [_ X / x ]_ A <-> ( Fun [_ X / x ]_ F /\ dom [_ X / x ]_ F = [_ X / x ]_ A ) )
12	9, 10, 11	3bitr4g 221	. 2 |- ( X e. V -> ( [. X / x ]. ( Fun F /\ dom F = A ) <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )
13	3, 12	bitrd 186	1 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   [.wsbc 2840   [_csb 2933   dom cdm 4438   Fun wfun 5009   Fn wfn 5010

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-fun 5017  df-fn 5018

This theorem is referenced by:  sbcfg  5160