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

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
Assertion
Ref Expression
rnmpt2ss
Distinct variable groups:   ,   ,,
Allowed substitution hints:   ()   (,)   (,)   (,)

Proof of Theorem rnmpt2ss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rnmpt2ss.1 . . . . 5
21rnmpt2 6397 . . . 4
32abeq2i 2570 . . 3
4 simpl 457 . . . . . 6
5 simpr 461 . . . . . 6
64, 5r19.29d2r 2986 . . . . 5
7 eleq1 2515 . . . . . . . 8
87biimparc 487 . . . . . . 7
98a1i 11 . . . . . 6
109rexlimivv 2940 . . . . 5
116, 10syl 16 . . . 4
1211ex 434 . . 3
133, 12syl5bi 217 . 2
1413ssrdv 3495 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1383   wcel 1804  wral 2793  wrex 2794   wss 3461   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
 Copyright terms: Public domain W3C validator