Theorem fnreseql 5744

Description: Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Assertion

Ref	Expression
fnreseql	|- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( ( F |` X ) = ( G |` X ) <-> X C_ dom ( F i^i G ) ) )

Proof of Theorem fnreseql

Step	Hyp	Ref	Expression
1		fnssres 5435	. . . 4 |- ( ( F Fn A /\ X C_ A ) -> ( F |` X ) Fn X )
2	1	3adant2 1040	. . 3 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( F |` X ) Fn X )
3		fnssres 5435	. . . 4 |- ( ( G Fn A /\ X C_ A ) -> ( G |` X ) Fn X )
4	3	3adant1 1039	. . 3 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( G |` X ) Fn X )
5		fneqeql 5742	. . 3 |- ( ( ( F |` X ) Fn X /\ ( G |` X ) Fn X ) -> ( ( F |` X ) = ( G |` X ) <-> dom ( ( F |` X ) i^i ( G |` X ) ) = X ) )
6	2, 4, 5	syl2anc 411	. 2 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( ( F |` X ) = ( G |` X ) <-> dom ( ( F |` X ) i^i ( G |` X ) ) = X ) )
7		resindir 5020	. . . . . 6 |- ( ( F i^i G ) |` X ) = ( ( F |` X ) i^i ( G |` X ) )
8	7	dmeqi 4923	. . . . 5 |- dom ( ( F i^i G ) |` X ) = dom ( ( F |` X ) i^i ( G |` X ) )
9		dmres 5025	. . . . 5 |- dom ( ( F i^i G ) |` X ) = ( X i^i dom ( F i^i G ) )
10	8, 9	eqtr3i 2252	. . . 4 |- dom ( ( F |` X ) i^i ( G |` X ) ) = ( X i^i dom ( F i^i G ) )
11	10	eqeq1i 2237	. . 3 |- ( dom ( ( F |` X ) i^i ( G |` X ) ) = X <-> ( X i^i dom ( F i^i G ) ) = X )
12		df-ss 3210	. . 3 |- ( X C_ dom ( F i^i G ) <-> ( X i^i dom ( F i^i G ) ) = X )
13	11, 12	bitr4i 187	. 2 |- ( dom ( ( F |` X ) i^i ( G |` X ) ) = X <-> X C_ dom ( F i^i G ) )
14	6, 13	bitrdi 196	1 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( ( F |` X ) = ( G |` X ) <-> X C_ dom ( F i^i G ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   i^i cin 3196   C_ wss 3197   dom cdm 4718   |` cres 4720   Fn wfn 5312

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by: (None)