Theorem dmiin 4908

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 3943	. . . 4 |- F/_ x |^|_ x e. A B
2	1	nfdm 4906	. . 3 |- F/_ x dom |^|_ x e. A B
3	2	ssiinf 3962	. 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 3965	. . 3 |- ( x e. A -> |^|_ x e. A B C_ B )
5		dmss 4861	. . 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 2552	1 |- dom |^|_ x e. A B C_ |^|_ x e. A dom B

Syntax hints:   e. wcel 2164   C_ wss 3153   |^|_ciin 3913   dom cdm 4659

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-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-iin 3915  df-br 4030  df-dm 4669

This theorem is referenced by: (None)