Theorem sbcfng 6042

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 5891	. . . 4 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2	1	a1i 11	. . 3 |- ( X e. V -> ( F Fn A <-> ( Fun F /\ dom F = A ) ) )
3	2	sbcbidv 3490	. 2 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [. X / x ]. ( Fun F /\ dom F = A ) ) )
4		sbcfung 5912	. . . 4 |- ( X e. V -> ( [. X / x ]. Fun F <-> Fun [_ X / x ]_ F ) )
5		sbceqg 3984	. . . . 5 |- ( X e. V -> ( [. X / x ]. dom F = A <-> [_ X / x ]_ dom F = [_ X / x ]_ A ) )
6		csbdm 5318	. . . . . 6 |- [_ X / x ]_ dom F = dom [_ X / x ]_ F
7	6	eqeq1i 2627	. . . . 5 |- ( [_ X / x ]_ dom F = [_ X / x ]_ A <-> dom [_ X / x ]_ F = [_ X / x ]_ A )
8	5, 7	syl6bb 276	. . . 4 |- ( X e. V -> ( [. X / x ]. dom F = A <-> dom [_ X / x ]_ F = [_ X / x ]_ A ) )
9	4, 8	anbi12d 747	. . 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 3478	. . 3 |- ( [. X / x ]. ( Fun F /\ dom F = A ) <-> ( [. X / x ]. Fun F /\ [. X / x ]. dom F = A ) )
11		df-fn 5891	. . 3 |- ( [_ X / x ]_ F Fn [_ X / x ]_ A <-> ( Fun [_ X / x ]_ F /\ dom [_ X / x ]_ F = [_ X / x ]_ A ) )
12	9, 10, 11	3bitr4g 303	. 2 |- ( X e. V -> ( [. X / x ]. ( Fun F /\ dom F = A ) <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )
13	3, 12	bitrd 268	1 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   [.wsbc 3435   [_csb 3533   dom cdm 5114   Fun wfun 5882   Fn wfn 5883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891

This theorem is referenced by:  sbcfg  6043