Theorem sbcfng 5470

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 5320	. . . 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 3087	. 2 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [. X / x ]. ( Fun F /\ dom F = A ) ) )
4		sbcfung 5341	. . . 4 |- ( X e. V -> ( [. X / x ]. Fun F <-> Fun [_ X / x ]_ F ) )
5		sbceqg 3140	. . . . 5 |- ( X e. V -> ( [. X / x ]. dom F = A <-> [_ X / x ]_ dom F = [_ X / x ]_ A ) )
6		csbdmg 4916	. . . . . 6 |- ( X e. V -> [_ X / x ]_ dom F = dom [_ X / x ]_ F )
7	6	eqeq1d 2238	. . . . 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 473	. . 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 3071	. . 3 |- ( [. X / x ]. ( Fun F /\ dom F = A ) <-> ( [. X / x ]. Fun F /\ [. X / x ]. dom F = A ) )
11		df-fn 5320	. . 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 1395   e. wcel 2200   [.wsbc 3028   [_csb 3124   dom cdm 4718   Fun wfun 5311   Fn wfn 5312

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-fun 5319  df-fn 5320

This theorem is referenced by:  sbcfg  5471