Theorem dmiin 4943

Description: Domain of an intersection. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
dmiin	|- dom |^|_ x e. A B C_ |^|_ x e. A dom B

Proof of Theorem dmiin

Step	Hyp	Ref	Expression
1		nfii1 3972	. . . 4 |- F/_ x |^|_ x e. A B
2	1	nfdm 4941	. . 3 |- F/_ x dom |^|_ x e. A B
3	2	ssiinf 3991	. 2 |- ( dom |^|_ x e. A B C_ |^|_ x e. A dom B <-> A. x e. A dom |^|_ x e. A B C_ dom B )
4		iinss2 3994	. . 3 |- ( x e. A -> |^|_ x e. A B C_ B )
5		dmss 4896	. . 3 |- ( |^|_ x e. A B C_ B -> dom |^|_ x e. A B C_ dom B )
6	4, 5	syl 14	. 2 |- ( x e. A -> dom |^|_ x e. A B C_ dom B )
7	3, 6	mprgbir 2566	1 |- dom |^|_ x e. A B C_ |^|_ x e. A dom B

Syntax hints:   e. wcel 2178   C_ wss 3174   |^|_ciin 3942   dom cdm 4693

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-iin 3944  df-br 4060  df-dm 4703

This theorem is referenced by: (None)