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Theorem rp-imass 38003
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 5021 . . 3  |-  ( R
" A )  =  ran  ( R  |`  A )
21sseq1i 3523 . 2  |-  ( ( R " A ) 
C_  B  <->  ran  ( R  |`  A )  C_  B
)
3 dmres 5304 . . . 4  |-  dom  ( R  |`  A )  =  ( A  i^i  dom  R )
4 inss1 3714 . . . 4  |-  ( A  i^i  dom  R )  C_  A
53, 4eqsstri 3529 . . 3  |-  dom  ( R  |`  A )  C_  A
65biantrur 506 . 2  |-  ( ran  ( R  |`  A ) 
C_  B  <->  ( dom  ( R  |`  A ) 
C_  A  /\  ran  ( R  |`  A ) 
C_  B ) )
7 relres 5311 . . . . 5  |-  Rel  ( R  |`  A )
8 relssdmrn 5534 . . . . 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 5117 . . . 4  |-  ( ( dom  ( R  |`  A )  C_  A  /\  ran  ( R  |`  A )  C_  B
)  ->  ( dom  ( R  |`  A )  X.  ran  ( R  |`  A ) )  C_  ( A  X.  B
) )
119, 10syl5ss 3510 . . 3  |-  ( ( dom  ( R  |`  A )  C_  A  /\  ran  ( R  |`  A )  C_  B
)  ->  ( R  |`  A )  C_  ( A  X.  B ) )
12 dmss 5212 . . . . 5  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  dom  ( R  |`  A )  C_  dom  ( A  X.  B
) )
13 dmxpss 5445 . . . . 5  |-  dom  ( A  X.  B )  C_  A
1412, 13syl6ss 3511 . . . 4  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  dom  ( R  |`  A )  C_  A
)
15 rnss 5241 . . . . 5  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  ran  ( R  |`  A )  C_  ran  ( A  X.  B
) )
16 rnxpss 5446 . . . . 5  |-  ran  ( A  X.  B )  C_  B
1715, 16syl6ss 3511 . . . 4  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  ran  ( R  |`  A )  C_  B
)
1814, 17jca 532 . . 3  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  ( dom  ( R  |`  A ) 
C_  A  /\  ran  ( R  |`  A ) 
C_  B ) )
1911, 18impbii 188 . 2  |-  ( ( dom  ( R  |`  A )  C_  A  /\  ran  ( R  |`  A )  C_  B
)  <->  ( R  |`  A )  C_  ( A  X.  B ) )
202, 6, 193bitri 271 1  |-  ( ( R " A ) 
C_  B  <->  ( R  |`  A )  C_  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    i^i cin 3470    C_ wss 3471    X. cxp 5006   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   Rel wrel 5013
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  df-res 5020  df-ima 5021
This theorem is referenced by:  dfhe2  38006
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