Theorem eqfnun 33516

Description: Two functions on A u. B are equal if and only if they have equal restrictions to both A and B. (Contributed by Jeff Madsen, 19-Jun-2011.)

Assertion

Ref	Expression
eqfnun	|- ( ( F Fn ( A u. B ) /\ G Fn ( A u. B ) ) -> ( F = G <-> ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) ) )

Proof of Theorem eqfnun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reseq1 5390	. . 3 |- ( F = G -> ( F |` A ) = ( G |` A ) )
2		reseq1 5390	. . 3 |- ( F = G -> ( F |` B ) = ( G |` B ) )
3	1, 2	jca 554	. 2 |- ( F = G -> ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) )
4		elun 3753	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
5		fveq1 6190	. . . . . . . . 9 |- ( ( F |` A ) = ( G |` A ) -> ( ( F |` A ) ` x ) = ( ( G |` A ) ` x ) )
6		fvres 6207	. . . . . . . . 9 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
7	5, 6	sylan9req 2677	. . . . . . . 8 |- ( ( ( F |` A ) = ( G |` A ) /\ x e. A ) -> ( ( G |` A ) ` x ) = ( F ` x ) )
8		fvres 6207	. . . . . . . . 9 |- ( x e. A -> ( ( G |` A ) ` x ) = ( G ` x ) )
9	8	adantl 482	. . . . . . . 8 |- ( ( ( F |` A ) = ( G |` A ) /\ x e. A ) -> ( ( G |` A ) ` x ) = ( G ` x ) )
10	7, 9	eqtr3d 2658	. . . . . . 7 |- ( ( ( F |` A ) = ( G |` A ) /\ x e. A ) -> ( F ` x ) = ( G ` x ) )
11	10	adantlr 751	. . . . . 6 |- ( ( ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) /\ x e. A ) -> ( F ` x ) = ( G ` x ) )
12		fveq1 6190	. . . . . . . . 9 |- ( ( F |` B ) = ( G |` B ) -> ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) )
13		fvres 6207	. . . . . . . . 9 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
14	12, 13	sylan9req 2677	. . . . . . . 8 |- ( ( ( F |` B ) = ( G |` B ) /\ x e. B ) -> ( ( G |` B ) ` x ) = ( F ` x ) )
15		fvres 6207	. . . . . . . . 9 |- ( x e. B -> ( ( G |` B ) ` x ) = ( G ` x ) )
16	15	adantl 482	. . . . . . . 8 |- ( ( ( F |` B ) = ( G |` B ) /\ x e. B ) -> ( ( G |` B ) ` x ) = ( G ` x ) )
17	14, 16	eqtr3d 2658	. . . . . . 7 |- ( ( ( F |` B ) = ( G |` B ) /\ x e. B ) -> ( F ` x ) = ( G ` x ) )
18	17	adantll 750	. . . . . 6 |- ( ( ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) /\ x e. B ) -> ( F ` x ) = ( G ` x ) )
19	11, 18	jaodan 826	. . . . 5 |- ( ( ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) /\ ( x e. A \/ x e. B ) ) -> ( F ` x ) = ( G ` x ) )
20	4, 19	sylan2b 492	. . . 4 |- ( ( ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) /\ x e. ( A u. B ) ) -> ( F ` x ) = ( G ` x ) )
21	20	ralrimiva 2966	. . 3 |- ( ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) -> A. x e. ( A u. B ) ( F ` x ) = ( G ` x ) )
22		eqfnfv 6311	. . 3 |- ( ( F Fn ( A u. B ) /\ G Fn ( A u. B ) ) -> ( F = G <-> A. x e. ( A u. B ) ( F ` x ) = ( G ` x ) ) )
23	21, 22	syl5ibr 236	. 2 |- ( ( F Fn ( A u. B ) /\ G Fn ( A u. B ) ) -> ( ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) -> F = G ) )
24	3, 23	impbid2 216	1 |- ( ( F Fn ( A u. B ) /\ G Fn ( A u. B ) ) -> ( F = G <-> ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   |` cres 5116   Fn wfn 5883   ` cfv 5888

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

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-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by: (None)