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Theorem rp-imass 36410
Description: If the  R-image of a class  A is a subclass of  B, then the restriction of  R to  A is a subset of the Cartesian product of  A and  B. (Contributed by Richard Penner, 24-Dec-2019.)
Assertion
Ref Expression
rp-imass  |-  ( ( R " A ) 
C_  B  <->  ( R  |`  A )  C_  ( A  X.  B ) )

Proof of Theorem rp-imass
StepHypRef Expression
1 df-ima 4865 . . 3  |-  ( R
" A )  =  ran  ( R  |`  A )
21sseq1i 3467 . 2  |-  ( ( R " A ) 
C_  B  <->  ran  ( R  |`  A )  C_  B
)
3 dmres 5143 . . . 4  |-  dom  ( R  |`  A )  =  ( A  i^i  dom  R )
4 inss1 3663 . . . 4  |-  ( A  i^i  dom  R )  C_  A
53, 4eqsstri 3473 . . 3  |-  dom  ( R  |`  A )  C_  A
65biantrur 513 . 2  |-  ( ran  ( R  |`  A ) 
C_  B  <->  ( dom  ( R  |`  A ) 
C_  A  /\  ran  ( R  |`  A ) 
C_  B ) )
7 relres 5150 . . . . 5  |-  Rel  ( R  |`  A )
8 relssdmrn 5374 . . . . 5  |-  ( Rel  ( R  |`  A )  ->  ( R  |`  A )  C_  ( dom  ( R  |`  A )  X.  ran  ( R  |`  A ) ) )
97, 8ax-mp 5 . . . 4  |-  ( R  |`  A )  C_  ( dom  ( R  |`  A )  X.  ran  ( R  |`  A ) )
10 xpss12 4958 . . . 4  |-  ( ( dom  ( R  |`  A )  C_  A  /\  ran  ( R  |`  A )  C_  B
)  ->  ( dom  ( R  |`  A )  X.  ran  ( R  |`  A ) )  C_  ( A  X.  B
) )
119, 10syl5ss 3454 . . 3  |-  ( ( dom  ( R  |`  A )  C_  A  /\  ran  ( R  |`  A )  C_  B
)  ->  ( R  |`  A )  C_  ( A  X.  B ) )
12 dmss 5052 . . . . 5  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  dom  ( R  |`  A )  C_  dom  ( A  X.  B
) )
13 dmxpss 5286 . . . . 5  |-  dom  ( A  X.  B )  C_  A
1412, 13syl6ss 3455 . . . 4  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  dom  ( R  |`  A )  C_  A
)
15 rnss 5081 . . . . 5  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  ran  ( R  |`  A )  C_  ran  ( A  X.  B
) )
16 rnxpss 5287 . . . . 5  |-  ran  ( A  X.  B )  C_  B
1715, 16syl6ss 3455 . . . 4  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  ran  ( R  |`  A )  C_  B
)
1814, 17jca 539 . . 3  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  ( dom  ( R  |`  A ) 
C_  A  /\  ran  ( R  |`  A ) 
C_  B ) )
1911, 18impbii 192 . 2  |-  ( ( dom  ( R  |`  A )  C_  A  /\  ran  ( R  |`  A )  C_  B
)  <->  ( R  |`  A )  C_  ( A  X.  B ) )
202, 6, 193bitri 279 1  |-  ( ( R " A ) 
C_  B  <->  ( R  |`  A )  C_  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    i^i cin 3414    C_ wss 3415    X. cxp 4850   dom cdm 4852   ran crn 4853    |` cres 4854   "cima 4855   Rel wrel 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-xp 4858  df-rel 4859  df-cnv 4860  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865
This theorem is referenced by:  dfhe2  36413
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