Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnmpt2ss Structured version   Unicode version

Theorem rnmpt2ss 27173
Description: The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)
Hypothesis
Ref Expression
rnmpt2ss.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
rnmpt2ss  |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  ->  ran  F  C_  D )
Distinct variable groups:    y, A    x, y, D
Allowed substitution hints:    A( x)    B( x, y)    C( x, y)    F( x, y)

Proof of Theorem rnmpt2ss
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 rnmpt2ss.1 . . . . 5  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
21rnmpt2 6387 . . . 4  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
32abeq2i 2587 . . 3  |-  ( z  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  z  =  C )
4 simpl 457 . . . . . 6  |-  ( ( A. x  e.  A  A. y  e.  B  C  e.  D  /\  E. x  e.  A  E. y  e.  B  z  =  C )  ->  A. x  e.  A  A. y  e.  B  C  e.  D )
5 simpr 461 . . . . . 6  |-  ( ( A. x  e.  A  A. y  e.  B  C  e.  D  /\  E. x  e.  A  E. y  e.  B  z  =  C )  ->  E. x  e.  A  E. y  e.  B  z  =  C )
64, 5r19.29d2r 2997 . . . . 5  |-  ( ( A. x  e.  A  A. y  e.  B  C  e.  D  /\  E. x  e.  A  E. y  e.  B  z  =  C )  ->  E. x  e.  A  E. y  e.  B  ( C  e.  D  /\  z  =  C ) )
7 eleq1 2532 . . . . . . . 8  |-  ( z  =  C  ->  (
z  e.  D  <->  C  e.  D ) )
87biimparc 487 . . . . . . 7  |-  ( ( C  e.  D  /\  z  =  C )  ->  z  e.  D )
98a1i 11 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ( C  e.  D  /\  z  =  C )  ->  z  e.  D ) )
109rexlimivv 2953 . . . . 5  |-  ( E. x  e.  A  E. y  e.  B  ( C  e.  D  /\  z  =  C )  ->  z  e.  D )
116, 10syl 16 . . . 4  |-  ( ( A. x  e.  A  A. y  e.  B  C  e.  D  /\  E. x  e.  A  E. y  e.  B  z  =  C )  ->  z  e.  D )
1211ex 434 . . 3  |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  ->  ( E. x  e.  A  E. y  e.  B  z  =  C  ->  z  e.  D ) )
133, 12syl5bi 217 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  ->  ( z  e.  ran  F  -> 
z  e.  D ) )
1413ssrdv 3503 1  |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  ->  ran  F  C_  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    C_ wss 3469   ran crn 4993    |-> cmpt2 6277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-cnv 5000  df-dm 5002  df-rn 5003  df-oprab 6279  df-mpt2 6280
This theorem is referenced by:  raddcn  27533  br2base  27866  sxbrsiga  27887
  Copyright terms: Public domain W3C validator