Theorem sbcfg 5402

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

Assertion

Ref	Expression
sbcfg	|- ( X e. V -> ( [. X / x ]. F : A --> B <-> [_ X / x ]_ F : [_ X / x ]_ A --> [_ X / x ]_ B ) )

Distinct variable groups:   x, V   x, X

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

Proof of Theorem sbcfg

Step	Hyp	Ref	Expression
1		df-f 5258	. . . 4 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2	1	a1i 9	. . 3 |- ( X e. V -> ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) ) )
3	2	sbcbidv 3044	. 2 |- ( X e. V -> ( [. X / x ]. F : A --> B <-> [. X / x ]. ( F Fn A /\ ran F C_ B ) ) )
4		sbcfng 5401	. . . 4 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )
5		sbcssg 3555	. . . . 5 |- ( X e. V -> ( [. X / x ]. ran F C_ B <-> [_ X / x ]_ ran F C_ [_ X / x ]_ B ) )
6		csbrng 5127	. . . . . 6 |- ( X e. V -> [_ X / x ]_ ran F = ran [_ X / x ]_ F )
7	6	sseq1d 3208	. . . . 5 |- ( X e. V -> ( [_ X / x ]_ ran F C_ [_ X / x ]_ B <-> ran [_ X / x ]_ F C_ [_ X / x ]_ B ) )
8	5, 7	bitrd 188	. . . 4 |- ( X e. V -> ( [. X / x ]. ran F C_ B <-> ran [_ X / x ]_ F C_ [_ X / x ]_ B ) )
9	4, 8	anbi12d 473	. . 3 |- ( X e. V -> ( ( [. X / x ]. F Fn A /\ [. X / x ]. ran F C_ B ) <-> ( [_ X / x ]_ F Fn [_ X / x ]_ A /\ ran [_ X / x ]_ F C_ [_ X / x ]_ B ) ) )
10		sbcan 3028	. . 3 |- ( [. X / x ]. ( F Fn A /\ ran F C_ B ) <-> ( [. X / x ]. F Fn A /\ [. X / x ]. ran F C_ B ) )
11		df-f 5258	. . 3 |- ( [_ X / x ]_ F : [_ X / x ]_ A --> [_ X / x ]_ B <-> ( [_ X / x ]_ F Fn [_ X / x ]_ A /\ ran [_ X / x ]_ F C_ [_ X / x ]_ B ) )
12	9, 10, 11	3bitr4g 223	. 2 |- ( X e. V -> ( [. X / x ]. ( F Fn A /\ ran F C_ B ) <-> [_ X / x ]_ F : [_ X / x ]_ A --> [_ X / x ]_ B ) )
13	3, 12	bitrd 188	1 |- ( X e. V -> ( [. X / x ]. F : A --> B <-> [_ X / x ]_ F : [_ X / x ]_ A --> [_ X / x ]_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2164   [.wsbc 2985   [_csb 3080   C_ wss 3153   ran crn 4660   Fn wfn 5249   -->wf 5250

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258

This theorem is referenced by:  csbwrdg  10943  ctiunctlemf  12595