Theorem 2elresin 5346

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 5338	. . . . . . . 8 |- ( ( F Fn A /\ <. x , y >. e. F ) -> x e. A )
2		fnop 5338	. . . . . . . 8 |- ( ( G Fn B /\ <. x , z >. e. G ) -> x e. B )
3	1, 2	anim12i 338	. . . . . . 7 |- ( ( ( F Fn A /\ <. x , y >. e. F ) /\ ( G Fn B /\ <. x , z >. e. G ) ) -> ( x e. A /\ x e. B ) )
4	3	an4s 588	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( <. x , y >. e. F /\ <. x , z >. e. G ) ) -> ( x e. A /\ x e. B ) )
5		elin 3333	. . . . . 6 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
6	4, 5	sylibr 134	. . . . 5 |- ( ( ( F Fn A /\ G Fn B ) /\ ( <. x , y >. e. F /\ <. x , z >. e. G ) ) -> x e. ( A i^i B ) )
7		vex 2755	. . . . . . . 8 |- y e. _V
8	7	opres 4934	. . . . . . 7 |- ( x e. ( A i^i B ) -> ( <. x , y >. e. ( F |` ( A i^i B ) ) <-> <. x , y >. e. F ) )
9		vex 2755	. . . . . . . 8 |- z e. _V
10	9	opres 4934	. . . . . . 7 |- ( x e. ( A i^i B ) -> ( <. x , z >. e. ( G |` ( A i^i B ) ) <-> <. x , z >. e. G ) )
11	8, 10	anbi12d 473	. . . . . 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 158	. . . . 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 115	. . 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 4949	. . . 4 |- ( F |` ( A i^i B ) ) C_ F
17	16	sseli 3166	. . 3 |- ( <. x , y >. e. ( F |` ( A i^i B ) ) -> <. x , y >. e. F )
18		resss 4949	. . . 4 |- ( G |` ( A i^i B ) ) C_ G
19	18	sseli 3166	. . 3 |- ( <. x , z >. e. ( G |` ( A i^i B ) ) -> <. x , z >. e. G )
20	17, 19	anim12i 338	. 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 142	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 104   <-> wb 105   e. wcel 2160   i^i cin 3143   <.cop 3610   |` cres 4646   Fn wfn 5230

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-dm 4654  df-res 4656  df-fun 5237  df-fn 5238

This theorem is referenced by: (None)