Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rp-imass Structured version   Unicode version

Theorem rp-imass 36279
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 4809 . . 3  |-  ( R
" A )  =  ran  ( R  |`  A )
21sseq1i 3431 . 2  |-  ( ( R " A ) 
C_  B  <->  ran  ( R  |`  A )  C_  B
)
3 dmres 5087 . . . 4  |-  dom  ( R  |`  A )  =  ( A  i^i  dom  R )
4 inss1 3625 . . . 4  |-  ( A  i^i  dom  R )  C_  A
53, 4eqsstri 3437 . . 3  |-  dom  ( R  |`  A )  C_  A
65biantrur 508 . 2  |-  ( ran  ( R  |`  A ) 
C_  B  <->  ( dom  ( R  |`  A ) 
C_  A  /\  ran  ( R  |`  A ) 
C_  B ) )
7 relres 5094 . . . . 5  |-  Rel  ( R  |`  A )
8 relssdmrn 5318 . . . . 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 4902 . . . 4  |-  ( ( dom  ( R  |`  A )  C_  A  /\  ran  ( R  |`  A )  C_  B
)  ->  ( dom  ( R  |`  A )  X.  ran  ( R  |`  A ) )  C_  ( A  X.  B
) )
119, 10syl5ss 3418 . . 3  |-  ( ( dom  ( R  |`  A )  C_  A  /\  ran  ( R  |`  A )  C_  B
)  ->  ( R  |`  A )  C_  ( A  X.  B ) )
12 dmss 4996 . . . . 5  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  dom  ( R  |`  A )  C_  dom  ( A  X.  B
) )
13 dmxpss 5230 . . . . 5  |-  dom  ( A  X.  B )  C_  A
1412, 13syl6ss 3419 . . . 4  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  dom  ( R  |`  A )  C_  A
)
15 rnss 5025 . . . . 5  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  ran  ( R  |`  A )  C_  ran  ( A  X.  B
) )
16 rnxpss 5231 . . . . 5  |-  ran  ( A  X.  B )  C_  B
1715, 16syl6ss 3419 . . . 4  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  ran  ( R  |`  A )  C_  B
)
1814, 17jca 534 . . 3  |-  ( ( R  |`  A )  C_  ( A  X.  B
)  ->  ( dom  ( R  |`  A ) 
C_  A  /\  ran  ( R  |`  A ) 
C_  B ) )
1911, 18impbii 190 . 2  |-  ( ( dom  ( R  |`  A )  C_  A  /\  ran  ( R  |`  A )  C_  B
)  <->  ( R  |`  A )  C_  ( A  X.  B ) )
202, 6, 193bitri 274 1  |-  ( ( R " A ) 
C_  B  <->  ( R  |`  A )  C_  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    i^i cin 3378    C_ wss 3379    X. cxp 4794   dom cdm 4796   ran crn 4797    |` cres 4798   "cima 4799   Rel wrel 4801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-xp 4802  df-rel 4803  df-cnv 4804  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809
This theorem is referenced by:  dfhe2  36282
  Copyright terms: Public domain W3C validator