Theorem rnin 4988

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 4986	. . . 4 |- `' ( A i^i B ) = ( `' A i^i `' B )
2	1	dmeqi 4780	. . 3 |- dom `' ( A i^i B ) = dom ( `' A i^i `' B )
3		dmin 4787	. . 3 |- dom ( `' A i^i `' B ) C_ ( dom `' A i^i dom `' B )
4	2, 3	eqsstri 3156	. 2 |- dom `' ( A i^i B ) C_ ( dom `' A i^i dom `' B )
5		df-rn 4590	. 2 |- ran ( A i^i B ) = dom `' ( A i^i B )
6		df-rn 4590	. . 3 |- ran A = dom `' A
7		df-rn 4590	. . 3 |- ran B = dom `' B
8	6, 7	ineq12i 3302	. 2 |- ( ran A i^i ran B ) = ( dom `' A i^i dom `' B )
9	4, 5, 8	3sstr4i 3165	1 |- ran ( A i^i B ) C_ ( ran A i^i ran B )

Syntax hints:   i^i cin 3097   C_ wss 3098   `'ccnv 4578   dom cdm 4579   ran crn 4580

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-xp 4585  df-rel 4586  df-cnv 4587  df-dm 4589  df-rn 4590

This theorem is referenced by:  inimass  4995