Theorem 2elresin 5234

Description: Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)

Assertion

Ref	Expression
2elresin	|- ( ( F Fn A /\ G Fn B ) -> ( ( <. x , y >. e. F /\ <. x , z >. e. G ) <-> ( <. x , y >. e. ( F |` ( A i^i B ) ) /\ <. x , z >. e. ( G |` ( A i^i B ) ) ) ) )

Proof of Theorem 2elresin

Step	Hyp	Ref	Expression
1		fnop 5226	. . . . . . . 8 |- ( ( F Fn A /\ <. x , y >. e. F ) -> x e. A )
2		fnop 5226	. . . . . . . 8 |- ( ( G Fn B /\ <. x , z >. e. G ) -> x e. B )
3	1, 2	anim12i 336	. . . . . . 7 |- ( ( ( F Fn A /\ <. x , y >. e. F ) /\ ( G Fn B /\ <. x , z >. e. G ) ) -> ( x e. A /\ x e. B ) )
4	3	an4s 577	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( <. x , y >. e. F /\ <. x , z >. e. G ) ) -> ( x e. A /\ x e. B ) )
5		elin 3259	. . . . . 6 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
6	4, 5	sylibr 133	. . . . 5 |- ( ( ( F Fn A /\ G Fn B ) /\ ( <. x , y >. e. F /\ <. x , z >. e. G ) ) -> x e. ( A i^i B ) )
7		vex 2689	. . . . . . . 8 |- y e. _V
8	7	opres 4828	. . . . . . 7 |- ( x e. ( A i^i B ) -> ( <. x , y >. e. ( F |` ( A i^i B ) ) <-> <. x , y >. e. F ) )
9		vex 2689	. . . . . . . 8 |- z e. _V
10	9	opres 4828	. . . . . . 7 |- ( x e. ( A i^i B ) -> ( <. x , z >. e. ( G |` ( A i^i B ) ) <-> <. x , z >. e. G ) )
11	8, 10	anbi12d 464	. . . . . 6 |- ( x e. ( A i^i B ) -> ( ( <. x , y >. e. ( F |` ( A i^i B ) ) /\ <. x , z >. e. ( G |` ( A i^i B ) ) ) <-> ( <. x , y >. e. F /\ <. x , z >. e. G ) ) )
12	11	biimprd 157	. . . . 5 |- ( x e. ( A i^i B ) -> ( ( <. x , y >. e. F /\ <. x , z >. e. G ) -> ( <. x , y >. e. ( F |` ( A i^i B ) ) /\ <. x , z >. e. ( G |` ( A i^i B ) ) ) ) )
13	6, 12	syl 14	. . . 4 |- ( ( ( F Fn A /\ G Fn B ) /\ ( <. x , y >. e. F /\ <. x , z >. e. G ) ) -> ( ( <. x , y >. e. F /\ <. x , z >. e. G ) -> ( <. x , y >. e. ( F |` ( A i^i B ) ) /\ <. x , z >. e. ( G |` ( A i^i B ) ) ) ) )
14	13	ex 114	. . 3 |- ( ( F Fn A /\ G Fn B ) -> ( ( <. x , y >. e. F /\ <. x , z >. e. G ) -> ( ( <. x , y >. e. F /\ <. x , z >. e. G ) -> ( <. x , y >. e. ( F |` ( A i^i B ) ) /\ <. x , z >. e. ( G |` ( A i^i B ) ) ) ) ) )
15	14	pm2.43d 50	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( ( <. x , y >. e. F /\ <. x , z >. e. G ) -> ( <. x , y >. e. ( F |` ( A i^i B ) ) /\ <. x , z >. e. ( G |` ( A i^i B ) ) ) ) )
16		resss 4843	. . . 4 |- ( F |` ( A i^i B ) ) C_ F
17	16	sseli 3093	. . 3 |- ( <. x , y >. e. ( F |` ( A i^i B ) ) -> <. x , y >. e. F )
18		resss 4843	. . . 4 |- ( G |` ( A i^i B ) ) C_ G
19	18	sseli 3093	. . 3 |- ( <. x , z >. e. ( G |` ( A i^i B ) ) -> <. x , z >. e. G )
20	17, 19	anim12i 336	. 2 |- ( ( <. x , y >. e. ( F |` ( A i^i B ) ) /\ <. x , z >. e. ( G |` ( A i^i B ) ) ) -> ( <. x , y >. e. F /\ <. x , z >. e. G ) )
21	15, 20	impbid1 141	1 |- ( ( F Fn A /\ G Fn B ) -> ( ( <. x , y >. e. F /\ <. x , z >. e. G ) <-> ( <. x , y >. e. ( F |` ( A i^i B ) ) /\ <. x , z >. e. ( G |` ( A i^i B ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   i^i cin 3070   <.cop 3530   |` cres 4541   Fn wfn 5118

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-dm 4549  df-res 4551  df-fun 5125  df-fn 5126

This theorem is referenced by: (None)