Theorem maprnin 29506

Description: Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.)

Hypotheses

Ref	Expression
maprnin.1	|- A e. _V
maprnin.2	|- B e. _V

Assertion

Ref	Expression
maprnin	|- ( ( B i^i C ) ^m A ) = { f e. ( B ^m A ) | ran f C_ C }

Distinct variable groups:   A, f   B, f   C, f

Proof of Theorem maprnin

Step	Hyp	Ref	Expression
1		ffn 6045	. . . . . 6 |- ( f : A --> B -> f Fn A )
2		df-f 5892	. . . . . . 7 |- ( f : A --> C <-> ( f Fn A /\ ran f C_ C ) )
3	2	baibr 945	. . . . . 6 |- ( f Fn A -> ( ran f C_ C <-> f : A --> C ) )
4	1, 3	syl 17	. . . . 5 |- ( f : A --> B -> ( ran f C_ C <-> f : A --> C ) )
5	4	pm5.32i 669	. . . 4 |- ( ( f : A --> B /\ ran f C_ C ) <-> ( f : A --> B /\ f : A --> C ) )
6		maprnin.2	. . . . . 6 |- B e. _V
7		maprnin.1	. . . . . 6 |- A e. _V
8	6, 7	elmap 7886	. . . . 5 |- ( f e. ( B ^m A ) <-> f : A --> B )
9	8	anbi1i 731	. . . 4 |- ( ( f e. ( B ^m A ) /\ ran f C_ C ) <-> ( f : A --> B /\ ran f C_ C ) )
10		fin 6085	. . . 4 |- ( f : A --> ( B i^i C ) <-> ( f : A --> B /\ f : A --> C ) )
11	5, 9, 10	3bitr4ri 293	. . 3 |- ( f : A --> ( B i^i C ) <-> ( f e. ( B ^m A ) /\ ran f C_ C ) )
12	11	abbii 2739	. 2 |- { f | f : A --> ( B i^i C ) } = { f | ( f e. ( B ^m A ) /\ ran f C_ C ) }
13	6	inex1 4799	. . 3 |- ( B i^i C ) e. _V
14	13, 7	mapval 7869	. 2 |- ( ( B i^i C ) ^m A ) = { f | f : A --> ( B i^i C ) }
15		df-rab 2921	. 2 |- { f e. ( B ^m A ) | ran f C_ C } = { f | ( f e. ( B ^m A ) /\ ran f C_ C ) }
16	12, 14, 15	3eqtr4i 2654	1 |- ( ( B i^i C ) ^m A ) = { f e. ( B ^m A ) | ran f C_ C }

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ran crn 5115   Fn wfn 5883   -->wf 5884  (class class class)co 6650   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by:  fpwrelmapffs  29509