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Theorem rnin 5422
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
StepHypRef Expression
1 cnvin 5420 . . . 4  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
21dmeqi 5214 . . 3  |-  dom  `' ( A  i^i  B )  =  dom  ( `' A  i^i  `' B
)
3 dmin 5220 . . 3  |-  dom  ( `' A  i^i  `' B
)  C_  ( dom  `' A  i^i  dom  `' B )
42, 3eqsstri 3529 . 2  |-  dom  `' ( A  i^i  B ) 
C_  ( dom  `' A  i^i  dom  `' B
)
5 df-rn 5019 . 2  |-  ran  ( A  i^i  B )  =  dom  `' ( A  i^i  B )
6 df-rn 5019 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 5019 . . 3  |-  ran  B  =  dom  `' B
86, 7ineq12i 3694 . 2  |-  ( ran 
A  i^i  ran  B )  =  ( dom  `' A  i^i  dom  `' B
)
94, 5, 83sstr4i 3538 1  |-  ran  ( A  i^i  B )  C_  ( ran  A  i^i  ran  B )
Colors of variables: wff setvar class
Syntax hints:    i^i cin 3470    C_ wss 3471   `'ccnv 5007   dom cdm 5008   ran crn 5009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019
This theorem is referenced by:  inimass  5429  restutop  20865
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