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Theorem rnin 5078
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 5076 . . . 4  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
21dmeqi 4868 . . 3  |-  dom  `' ( A  i^i  B )  =  dom  ( `' A  i^i  `' B
)
3 dmin 4874 . . 3  |-  dom  ( `' A  i^i  `' B
)  C_  ( dom  `'  A  i^i  dom  `'  B )
42, 3eqsstri 3183 . 2  |-  dom  `' ( A  i^i  B ) 
C_  ( dom  `'  A  i^i  dom  `'  B
)
5 df-rn 4680 . 2  |-  ran  (  A  i^i  B )  =  dom  `' ( A  i^i  B )
6 df-rn 4680 . . 3  |-  ran  A  =  dom  `'  A
7 df-rn 4680 . . 3  |-  ran  B  =  dom  `'  B
86, 7ineq12i 3343 . 2  |-  ( ran 
A  i^i  ran  B )  =  ( dom  `'  A  i^i  dom  `'  B
)
94, 5, 83sstr4i 3192 1  |-  ran  (  A  i^i  B )  C_  ( ran  A  i^i  ran  B )
Colors of variables: wff set class
Syntax hints:    i^i cin 3126    C_ wss 3127   `'ccnv 4660   dom cdm 4661   ran crn 4662
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-rel 4676  df-cnv 4677  df-dm 4679  df-rn 4680
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