Theorem sbcfg 5045

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 4906	. . . 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 2817	. 2 |- ( X e. V -> ( [. X / x ]. F : A --> B <-> [. X / x ]. ( F Fn A /\ ran F C_ B ) ) )
4		sbcfng 5044	. . . 4 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )
5		sbcssg 3330	. . . . 5 |- ( X e. V -> ( [. X / x ]. ran F C_ B <-> [_ X / x ]_ ran F C_ [_ X / x ]_ B ) )
6		csbrng 4782	. . . . . 6 |- ( X e. V -> [_ X / x ]_ ran F = ran [_ X / x ]_ F )
7	6	sseq1d 2972	. . . . 5 |- ( X e. V -> ( [_ X / x ]_ ran F C_ [_ X / x ]_ B <-> ran [_ X / x ]_ F C_ [_ X / x ]_ B ) )
8	5, 7	bitrd 177	. . . 4 |- ( X e. V -> ( [. X / x ]. ran F C_ B <-> ran [_ X / x ]_ F C_ [_ X / x ]_ B ) )
9	4, 8	anbi12d 442	. . 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 2805	. . 3 |- ( [. X / x ]. ( F Fn A /\ ran F C_ B ) <-> ( [. X / x ]. F Fn A /\ [. X / x ]. ran F C_ B ) )
11		df-f 4906	. . 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 212	. 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 177	1 |- ( X e. V -> ( [. X / x ]. F : A --> B <-> [_ X / x ]_ F : [_ X / x ]_ A --> [_ X / x ]_ B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   [.wsbc 2764   [_csb 2852   C_ wss 2917   ran crn 4346   Fn wfn 4897   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906

This theorem is referenced by: (None)