Theorem rnin 4733

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 4731	. . . 4 |- `' ( A i^i B ) = ( `' A i^i `' B )
2	1	dmeqi 4536	. . 3 |- dom `' ( A i^i B ) = dom ( `' A i^i `' B )
3		dmin 4543	. . 3 |- dom ( `' A i^i `' B ) C_ ( dom `' A i^i dom `' B )
4	2, 3	eqsstri 2975	. 2 |- dom `' ( A i^i B ) C_ ( dom `' A i^i dom `' B )
5		df-rn 4356	. 2 |- ran ( A i^i B ) = dom `' ( A i^i B )
6		df-rn 4356	. . 3 |- ran A = dom `' A
7		df-rn 4356	. . 3 |- ran B = dom `' B
8	6, 7	ineq12i 3136	. 2 |- ( ran A i^i ran B ) = ( dom `' A i^i dom `' B )
9	4, 5, 8	3sstr4i 2984	1 |- ran ( A i^i B ) C_ ( ran A i^i ran B )

Syntax hints:   i^i cin 2916   C_ wss 2917   `'ccnv 4344   dom cdm 4345   ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  inimass  4740