Theorem inimass 5047

Description: The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
inimass	|- ( ( A i^i B ) " C ) C_ ( ( A " C ) i^i ( B " C ) )

Proof of Theorem inimass

Step	Hyp	Ref	Expression
1		rnin 5040	. 2 |- ran ( ( A |` C ) i^i ( B |` C ) ) C_ ( ran ( A |` C ) i^i ran ( B |` C ) )
2		df-ima 4641	. . 3 |- ( ( A i^i B ) " C ) = ran ( ( A i^i B ) |` C )
3		resindir 4925	. . . 4 |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )
4	3	rneqi 4857	. . 3 |- ran ( ( A i^i B ) |` C ) = ran ( ( A |` C ) i^i ( B |` C ) )
5	2, 4	eqtri 2198	. 2 |- ( ( A i^i B ) " C ) = ran ( ( A |` C ) i^i ( B |` C ) )
6		df-ima 4641	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4641	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	ineq12i 3336	. 2 |- ( ( A " C ) i^i ( B " C ) ) = ( ran ( A |` C ) i^i ran ( B |` C ) )
9	1, 5, 8	3sstr4i 3198	1 |- ( ( A i^i B ) " C ) C_ ( ( A " C ) i^i ( B " C ) )

Syntax hints:   i^i cin 3130   C_ wss 3131   ran crn 4629   |` cres 4630   "cima 4631

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641

This theorem is referenced by: (None)