Theorem inimass 5027

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 5020	. 2 |- ran ( ( A |` C ) i^i ( B |` C ) ) C_ ( ran ( A |` C ) i^i ran ( B |` C ) )
2		df-ima 4624	. . 3 |- ( ( A i^i B ) " C ) = ran ( ( A i^i B ) |` C )
3		resindir 4907	. . . 4 |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )
4	3	rneqi 4839	. . 3 |- ran ( ( A i^i B ) |` C ) = ran ( ( A |` C ) i^i ( B |` C ) )
5	2, 4	eqtri 2191	. 2 |- ( ( A i^i B ) " C ) = ran ( ( A |` C ) i^i ( B |` C ) )
6		df-ima 4624	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4624	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	ineq12i 3326	. 2 |- ( ( A " C ) i^i ( B " C ) ) = ( ran ( A |` C ) i^i ran ( B |` C ) )
9	1, 5, 8	3sstr4i 3188	1 |- ( ( A i^i B ) " C ) C_ ( ( A " C ) i^i ( B " C ) )

Syntax hints:   i^i cin 3120   C_ wss 3121   ran crn 4612   |` cres 4613   "cima 4614

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by: (None)