Theorem sbcfg 5481

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 5330	. . . 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 3090	. 2 |- ( X e. V -> ( [. X / x ]. F : A --> B <-> [. X / x ]. ( F Fn A /\ ran F C_ B ) ) )
4		sbcfng 5480	. . . 4 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )
5		sbcssg 3603	. . . . 5 |- ( X e. V -> ( [. X / x ]. ran F C_ B <-> [_ X / x ]_ ran F C_ [_ X / x ]_ B ) )
6		csbrng 5198	. . . . . 6 |- ( X e. V -> [_ X / x ]_ ran F = ran [_ X / x ]_ F )
7	6	sseq1d 3256	. . . . 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 3074	. . 3 |- ( [. X / x ]. ( F Fn A /\ ran F C_ B ) <-> ( [. X / x ]. F Fn A /\ [. X / x ]. ran F C_ B ) )
11		df-f 5330	. . 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 2202   [.wsbc 3031   [_csb 3127   C_ wss 3200   ran crn 4726   Fn wfn 5321   -->wf 5322

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330

This theorem is referenced by:  csbwrdg  11142  ctiunctlemf  13058