Theorem inimass 5020

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 5013	. 2 |- ran ( ( A |` C ) i^i ( B |` C ) ) C_ ( ran ( A |` C ) i^i ran ( B |` C ) )
2		df-ima 4617	. . 3 |- ( ( A i^i B ) " C ) = ran ( ( A i^i B ) |` C )
3		resindir 4900	. . . 4 |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )
4	3	rneqi 4832	. . 3 |- ran ( ( A i^i B ) |` C ) = ran ( ( A |` C ) i^i ( B |` C ) )
5	2, 4	eqtri 2186	. 2 |- ( ( A i^i B ) " C ) = ran ( ( A |` C ) i^i ( B |` C ) )
6		df-ima 4617	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4617	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	ineq12i 3321	. 2 |- ( ( A " C ) i^i ( B " C ) ) = ( ran ( A |` C ) i^i ran ( B |` C ) )
9	1, 5, 8	3sstr4i 3183	1 |- ( ( A i^i B ) " C ) C_ ( ( A " C ) i^i ( B " C ) )

Syntax hints:   i^i cin 3115   C_ wss 3116   ran crn 4605   |` cres 4606   "cima 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by: (None)