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Theorem iso0 27404
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 5671 . 2  |-  (/) : (/) -1-1-onto-> (/)
2 ral0 3692 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  ( x R y  <-> 
( (/) `  x ) S ( (/) `  y
) )
3 df-isom 5422 . 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 2666   (/)c0 3588   class class class wbr 4172   -1-1-onto->wf1o 5412   ` cfv 5413    Isom wiso 5414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-isom 5422
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