Theorem 2elresin 5081

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 5073	. . . . . . . 8 |- ( ( F Fn A /\ <. x , y >. e. F ) -> x e. A )
2		fnop 5073	. . . . . . . 8 |- ( ( G Fn B /\ <. x , z >. e. G ) -> x e. B )
3	1, 2	anim12i 331	. . . . . . 7 |- ( ( ( F Fn A /\ <. x , y >. e. F ) /\ ( G Fn B /\ <. x , z >. e. G ) ) -> ( x e. A /\ x e. B ) )
4	3	an4s 553	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( <. x , y >. e. F /\ <. x , z >. e. G ) ) -> ( x e. A /\ x e. B ) )
5		elin 3169	. . . . . 6 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
6	4, 5	sylibr 132	. . . . 5 |- ( ( ( F Fn A /\ G Fn B ) /\ ( <. x , y >. e. F /\ <. x , z >. e. G ) ) -> x e. ( A i^i B ) )
7		vex 2617	. . . . . . . 8 |- y e. _V
8	7	opres 4683	. . . . . . 7 |- ( x e. ( A i^i B ) -> ( <. x , y >. e. ( F |` ( A i^i B ) ) <-> <. x , y >. e. F ) )
9		vex 2617	. . . . . . . 8 |- z e. _V
10	9	opres 4683	. . . . . . 7 |- ( x e. ( A i^i B ) -> ( <. x , z >. e. ( G |` ( A i^i B ) ) <-> <. x , z >. e. G ) )
11	8, 10	anbi12d 457	. . . . . 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 156	. . . . 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 113	. . 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 49	. 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 4697	. . . 4 |- ( F |` ( A i^i B ) ) C_ F
17	16	sseli 3008	. . 3 |- ( <. x , y >. e. ( F |` ( A i^i B ) ) -> <. x , y >. e. F )
18		resss 4697	. . . 4 |- ( G |` ( A i^i B ) ) C_ G
19	18	sseli 3008	. . 3 |- ( <. x , z >. e. ( G |` ( A i^i B ) ) -> <. x , z >. e. G )
20	17, 19	anim12i 331	. 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 140	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 102   <-> wb 103   e. wcel 1436   i^i cin 2985   <.cop 3428   |` cres 4406   Fn wfn 4967

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-opab 3869  df-xp 4410  df-rel 4411  df-dm 4414  df-res 4416  df-fun 4974  df-fn 4975

This theorem is referenced by: (None)