Theorem sbcfng 5497

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 5346	. . . 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 3100	. 2 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [. X / x ]. ( Fun F /\ dom F = A ) ) )
4		sbcfung 5367	. . . 4 |- ( X e. V -> ( [. X / x ]. Fun F <-> Fun [_ X / x ]_ F ) )
5		sbceqg 3153	. . . . 5 |- ( X e. V -> ( [. X / x ]. dom F = A <-> [_ X / x ]_ dom F = [_ X / x ]_ A ) )
6		csbdmg 4941	. . . . . 6 |- ( X e. V -> [_ X / x ]_ dom F = dom [_ X / x ]_ F )
7	6	eqeq1d 2241	. . . . 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 3084	. . 3 |- ( [. X / x ]. ( Fun F /\ dom F = A ) <-> ( [. X / x ]. Fun F /\ [. X / x ]. dom F = A ) )
11		df-fn 5346	. . 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 1398   e. wcel 2203   [.wsbc 3041   [_csb 3137   dom cdm 4740   Fun wfun 5337   Fn wfn 5338

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-br 4103  df-opab 4165  df-id 4405  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-fun 5345  df-fn 5346

This theorem is referenced by:  sbcfg  5498