Theorem inimass 4790

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 4783	. 2 |- ran ( ( A |` C ) i^i ( B |` C ) ) C_ ( ran ( A |` C ) i^i ran ( B |` C ) )
2		df-ima 4404	. . 3 |- ( ( A i^i B ) " C ) = ran ( ( A i^i B ) |` C )
3		resindir 4676	. . . 4 |- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )
4	3	rneqi 4610	. . 3 |- ran ( ( A i^i B ) |` C ) = ran ( ( A |` C ) i^i ( B |` C ) )
5	2, 4	eqtri 2103	. 2 |- ( ( A i^i B ) " C ) = ran ( ( A |` C ) i^i ( B |` C ) )
6		df-ima 4404	. . 3 |- ( A " C ) = ran ( A |` C )
7		df-ima 4404	. . 3 |- ( B " C ) = ran ( B |` C )
8	6, 7	ineq12i 3181	. 2 |- ( ( A " C ) i^i ( B " C ) ) = ( ran ( A |` C ) i^i ran ( B |` C ) )
9	1, 5, 8	3sstr4i 3047	1 |- ( ( A i^i B ) " C ) C_ ( ( A " C ) i^i ( B " C ) )

Syntax hints:   i^i cin 2981   C_ wss 2982   ran crn 4392   |` cres 4393   "cima 4394

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404

This theorem is referenced by: (None)