Theorem dmiin 5369

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 4551	. . . 4 |- F/_ x |^|_ x e. A B
2	1	nfdm 5367	. . 3 |- F/_ x dom |^|_ x e. A B
3	2	ssiinf 4569	. 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 4572	. . 3 |- ( x e. A -> |^|_ x e. A B C_ B )
5		dmss 5323	. . 3 |- ( |^|_ x e. A B C_ B -> dom |^|_ x e. A B C_ dom B )
6	4, 5	syl 17	. 2 |- ( x e. A -> dom |^|_ x e. A B C_ dom B )
7	3, 6	mprgbir 2927	1 |- dom |^|_ x e. A B C_ |^|_ x e. A dom B

Syntax hints:   e. wcel 1990   C_ wss 3574   |^|_ciin 4521   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-ral 2917  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-iin 4523  df-br 4654  df-dm 5124

This theorem is referenced by: (None)