Theorem rnin 5040

Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Assertion

Ref	Expression
rnin	|- ran ( A i^i B ) C_ ( ran A i^i ran B )

Proof of Theorem rnin

Step	Hyp	Ref	Expression
1		cnvin 5038	. . . 4 |- `' ( A i^i B ) = ( `' A i^i `' B )
2	1	dmeqi 4830	. . 3 |- dom `' ( A i^i B ) = dom ( `' A i^i `' B )
3		dmin 4837	. . 3 |- dom ( `' A i^i `' B ) C_ ( dom `' A i^i dom `' B )
4	2, 3	eqsstri 3189	. 2 |- dom `' ( A i^i B ) C_ ( dom `' A i^i dom `' B )
5		df-rn 4639	. 2 |- ran ( A i^i B ) = dom `' ( A i^i B )
6		df-rn 4639	. . 3 |- ran A = dom `' A
7		df-rn 4639	. . 3 |- ran B = dom `' B
8	6, 7	ineq12i 3336	. 2 |- ( ran A i^i ran B ) = ( dom `' A i^i dom `' B )
9	4, 5, 8	3sstr4i 3198	1 |- ran ( A i^i B ) C_ ( ran A i^i ran B )

Syntax hints:   i^i cin 3130   C_ wss 3131   `'ccnv 4627   dom cdm 4628   ran crn 4629

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

This theorem is referenced by:  inimass  5047