Theorem sbcfg 5356

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 5212	. . . 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 3019	. 2 |- ( X e. V -> ( [. X / x ]. F : A --> B <-> [. X / x ]. ( F Fn A /\ ran F C_ B ) ) )
4		sbcfng 5355	. . . 4 |- ( X e. V -> ( [. X / x ]. F Fn A <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )
5		sbcssg 3530	. . . . 5 |- ( X e. V -> ( [. X / x ]. ran F C_ B <-> [_ X / x ]_ ran F C_ [_ X / x ]_ B ) )
6		csbrng 5082	. . . . . 6 |- ( X e. V -> [_ X / x ]_ ran F = ran [_ X / x ]_ F )
7	6	sseq1d 3182	. . . . 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 3003	. . 3 |- ( [. X / x ]. ( F Fn A /\ ran F C_ B ) <-> ( [. X / x ]. F Fn A /\ [. X / x ]. ran F C_ B ) )
11		df-f 5212	. . 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 2146   [.wsbc 2960   [_csb 3055   C_ wss 3127   ran crn 4621   Fn wfn 5203   -->wf 5204

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-id 4287  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-fun 5210  df-fn 5211  df-f 5212

This theorem is referenced by:  ctiunctlemf  12404