Theorem fnreseql 5757

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 5445	. . . 4 |- ( ( F Fn A /\ X C_ A ) -> ( F |` X ) Fn X )
2	1	3adant2 1042	. . 3 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( F |` X ) Fn X )
3		fnssres 5445	. . . 4 |- ( ( G Fn A /\ X C_ A ) -> ( G |` X ) Fn X )
4	3	3adant1 1041	. . 3 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( G |` X ) Fn X )
5		fneqeql 5755	. . 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 5029	. . . . . 6 |- ( ( F i^i G ) |` X ) = ( ( F |` X ) i^i ( G |` X ) )
8	7	dmeqi 4932	. . . . 5 |- dom ( ( F i^i G ) |` X ) = dom ( ( F |` X ) i^i ( G |` X ) )
9		dmres 5034	. . . . 5 |- dom ( ( F i^i G ) |` X ) = ( X i^i dom ( F i^i G ) )
10	8, 9	eqtr3i 2254	. . . 4 |- dom ( ( F |` X ) i^i ( G |` X ) ) = ( X i^i dom ( F i^i G ) )
11	10	eqeq1i 2239	. . 3 |- ( dom ( ( F |` X ) i^i ( G |` X ) ) = X <-> ( X i^i dom ( F i^i G ) ) = X )
12		df-ss 3213	. . 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 1004   = wceq 1397   i^i cin 3199   C_ wss 3200   dom cdm 4725   |` cres 4727   Fn wfn 5321

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by: (None)