Theorem dmin 5332

Description: The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Assertion

Ref	Expression
dmin	|- dom ( A i^i B ) C_ ( dom A i^i dom B )

Proof of Theorem dmin

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.40 1797	. . 3 |- ( E. y ( <. x , y >. e. A /\ <. x , y >. e. B ) -> ( E. y <. x , y >. e. A /\ E. y <. x , y >. e. B ) )
2		vex 3203	. . . . 5 |- x e. _V
3	2	eldm2 5322	. . . 4 |- ( x e. dom ( A i^i B ) <-> E. y <. x , y >. e. ( A i^i B ) )
4		elin 3796	. . . . 5 |- ( <. x , y >. e. ( A i^i B ) <-> ( <. x , y >. e. A /\ <. x , y >. e. B ) )
5	4	exbii 1774	. . . 4 |- ( E. y <. x , y >. e. ( A i^i B ) <-> E. y ( <. x , y >. e. A /\ <. x , y >. e. B ) )
6	3, 5	bitri 264	. . 3 |- ( x e. dom ( A i^i B ) <-> E. y ( <. x , y >. e. A /\ <. x , y >. e. B ) )
7		elin 3796	. . . 4 |- ( x e. ( dom A i^i dom B ) <-> ( x e. dom A /\ x e. dom B ) )
8	2	eldm2 5322	. . . . 5 |- ( x e. dom A <-> E. y <. x , y >. e. A )
9	2	eldm2 5322	. . . . 5 |- ( x e. dom B <-> E. y <. x , y >. e. B )
10	8, 9	anbi12i 733	. . . 4 |- ( ( x e. dom A /\ x e. dom B ) <-> ( E. y <. x , y >. e. A /\ E. y <. x , y >. e. B ) )
11	7, 10	bitri 264	. . 3 |- ( x e. ( dom A i^i dom B ) <-> ( E. y <. x , y >. e. A /\ E. y <. x , y >. e. B ) )
12	1, 6, 11	3imtr4i 281	. 2 |- ( x e. dom ( A i^i B ) -> x e. ( dom A i^i dom B ) )
13	12	ssriv 3607	1 |- dom ( A i^i B ) C_ ( dom A i^i dom B )

Syntax hints:   /\ wa 384   E.wex 1704   e. wcel 1990   i^i cin 3573   C_ wss 3574   <.cop 4183   dom cdm 5114

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  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-br 4654  df-dm 5124

This theorem is referenced by:  rnin  5542  psssdm2  17215  hauseqcn  29941