Theorem fnreseql 5277

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 5012	. . . 4 |- ( ( F Fn A /\ X C_ A ) -> ( F |` X ) Fn X )
2	1	3adant2 923	. . 3 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( F |` X ) Fn X )
3		fnssres 5012	. . . 4 |- ( ( G Fn A /\ X C_ A ) -> ( G |` X ) Fn X )
4	3	3adant1 922	. . 3 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( G |` X ) Fn X )
5		fneqeql 5275	. . 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 391	. 2 |- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( ( F |` X ) = ( G |` X ) <-> dom ( ( F |` X ) i^i ( G |` X ) ) = X ) )
7		resindir 4628	. . . . . 6 |- ( ( F i^i G ) |` X ) = ( ( F |` X ) i^i ( G |` X ) )
8	7	dmeqi 4536	. . . . 5 |- dom ( ( F i^i G ) |` X ) = dom ( ( F |` X ) i^i ( G |` X ) )
9		dmres 4632	. . . . 5 |- dom ( ( F i^i G ) |` X ) = ( X i^i dom ( F i^i G ) )
10	8, 9	eqtr3i 2062	. . . 4 |- dom ( ( F |` X ) i^i ( G |` X ) ) = ( X i^i dom ( F i^i G ) )
11	10	eqeq1i 2047	. . 3 |- ( dom ( ( F |` X ) i^i ( G |` X ) ) = X <-> ( X i^i dom ( F i^i G ) ) = X )
12		df-ss 2931	. . 3 |- ( X C_ dom ( F i^i G ) <-> ( X i^i dom ( F i^i G ) ) = X )
13	11, 12	bitr4i 176	. 2 |- ( dom ( ( F |` X ) i^i ( G |` X ) ) = X <-> X C_ dom ( F i^i G ) )
14	6, 13	syl6bb 185	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 98   /\ w3a 885   = wceq 1243   i^i cin 2916   C_ wss 2917   dom cdm 4345   |` cres 4347   Fn wfn 4897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by: (None)