Theorem inimass 4913

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 4906	. 2 |- ran ( ( A |` C ) i^i ( B |` C ) ) C_ ( ran ( A |` C ) i^i ran ( B |` C ) )
2		df-ima 4512	. . 3 |- ( ( A i^i B ) " C ) = ran ( ( A i^i B ) |` C )
3		resindir 4793	. . . 4 |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )
4	3	rneqi 4727	. . 3 |- ran ( ( A i^i B ) |` C ) = ran ( ( A |` C ) i^i ( B |` C ) )
5	2, 4	eqtri 2135	. 2 |- ( ( A i^i B ) " C ) = ran ( ( A |` C ) i^i ( B |` C ) )
6		df-ima 4512	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4512	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	ineq12i 3241	. 2 |- ( ( A " C ) i^i ( B " C ) ) = ( ran ( A |` C ) i^i ran ( B |` C ) )
9	1, 5, 8	3sstr4i 3104	1 |- ( ( A i^i B ) " C ) C_ ( ( A " C ) i^i ( B " C ) )

Syntax hints:   i^i cin 3036   C_ wss 3037   ran crn 4500   |` cres 4501   "cima 4502

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-cnv 4507  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512

This theorem is referenced by: (None)