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Theorem rnmpt2ss 27493
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 6397 . . . 4  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
32abeq2i 2570 . . 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 2986 . . . . 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 2515 . . . . . . . 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 2940 . . . . 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 3495 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 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    C_ wss 3461   ran crn 4990    |-> cmpt2 6283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-cnv 4997  df-dm 4999  df-rn 5000  df-oprab 6285  df-mpt2 6286
This theorem is referenced by:  raddcn  27889  br2base  28218  sxbrsiga  28239
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