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Theorem iso0 27446
Description: The empty set is an  R ,  S isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
Assertion
Ref Expression
iso0  |-  (/)  Isom  R ,  S  ( (/) ,  (/) )

Proof of Theorem iso0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1o0 5698 . 2  |-  (/) : (/) -1-1-onto-> (/)
2 ral0 3719 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  ( x R y  <-> 
( (/) `  x ) S ( (/) `  y
) )
3 df-isom 5449 . 2  |-  ( (/)  Isom 
R ,  S  (
(/) ,  (/) )  <->  ( (/) : (/) -1-1-onto-> (/)  /\  A. x  e.  (/)  A. y  e.  (/)  ( x R y  <->  ( (/) `  x
) S ( (/) `  y ) ) ) )
41, 2, 3mpbir2an 887 1  |-  (/)  Isom  R ,  S  ( (/) ,  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wral 2692   (/)c0 3615   class class class wbr 4199   -1-1-onto->wf1o 5439   ` cfv 5440    Isom wiso 5441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-br 4200  df-opab 4254  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-isom 5449
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