Theorem fnreseql 5523

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 5231	. . . 4 |- ( ( F Fn A /\ X C_ A ) -> ( F |` X ) Fn X )
2	1	3adant2 1000	. . 3 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( F |` X ) Fn X )
3		fnssres 5231	. . . 4 |- ( ( G Fn A /\ X C_ A ) -> ( G |` X ) Fn X )
4	3	3adant1 999	. . 3 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( G |` X ) Fn X )
5		fneqeql 5521	. . 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 408	. 2 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( ( F |` X ) = ( G |` X ) <-> dom ( ( F |` X ) i^i ( G |` X ) ) = X ) )
7		resindir 4830	. . . . . 6 |- ( ( F i^i G ) |` X ) = ( ( F |` X ) i^i ( G |` X ) )
8	7	dmeqi 4735	. . . . 5 |- dom ( ( F i^i G ) |` X ) = dom ( ( F |` X ) i^i ( G |` X ) )
9		dmres 4835	. . . . 5 |- dom ( ( F i^i G ) |` X ) = ( X i^i dom ( F i^i G ) )
10	8, 9	eqtr3i 2160	. . . 4 |- dom ( ( F |` X ) i^i ( G |` X ) ) = ( X i^i dom ( F i^i G ) )
11	10	eqeq1i 2145	. . 3 |- ( dom ( ( F |` X ) i^i ( G |` X ) ) = X <-> ( X i^i dom ( F i^i G ) ) = X )
12		df-ss 3079	. . 3 |- ( X C_ dom ( F i^i G ) <-> ( X i^i dom ( F i^i G ) ) = X )
13	11, 12	bitr4i 186	. 2 |- ( dom ( ( F |` X ) i^i ( G |` X ) ) = X <-> X C_ dom ( F i^i G ) )
14	6, 13	syl6bb 195	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 104   /\ w3a 962   = wceq 1331   i^i cin 3065   C_ wss 3066   dom cdm 4534   |` cres 4536   Fn wfn 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by: (None)