Theorem sbcfng 5529

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 5378	. . . 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 3110	. 2 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [. X / x ]. ( Fun F /\ dom F = A ) ) )
4		sbcfung 5399	. . . 4 |- ( X e. V -> ( [. X / x ]. Fun F <-> Fun [_ X / x ]_ F ) )
5		sbceqg 3163	. . . . 5 |- ( X e. V -> ( [. X / x ]. dom F = A <-> [_ X / x ]_ dom F = [_ X / x ]_ A ) )
6		csbdmg 4973	. . . . . 6 |- ( X e. V -> [_ X / x ]_ dom F = dom [_ X / x ]_ F )
7	6	eqeq1d 2247	. . . . 5 |- ( X e. V -> ( [_ X / x ]_ dom F = [_ X / x ]_ A <-> dom [_ X / x ]_ F = [_ X / x ]_ A ) )
8	5, 7	bitrd 188	. . . 4 |- ( X e. V -> ( [. X / x ]. dom F = A <-> dom [_ X / x ]_ F = [_ X / x ]_ A ) )
9	4, 8	anbi12d 477	. . 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 3094	. . 3 |- ( [. X / x ]. ( Fun F /\ dom F = A ) <-> ( [. X / x ]. Fun F /\ [. X / x ]. dom F = A ) )
11		df-fn 5378	. . 3 |- ( [_ X / x ]_ F Fn [_ X / x ]_ A <-> ( Fun [_ X / x ]_ F /\ dom [_ X / x ]_ F = [_ X / x ]_ A ) )
12	9, 10, 11	3bitr4g 223	. 2 |- ( X e. V -> ( [. X / x ]. ( Fun F /\ dom F = A ) <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )
13	3, 12	bitrd 188	1 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   [.wsbc 3051   [_csb 3147   dom cdm 4772   Fun wfun 5369   Fn wfn 5370

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-id 4436  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-fun 5377  df-fn 5378

This theorem is referenced by:  sbcfg  5530