Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqrelrd2 Structured version   Unicode version

Theorem eqrelrd2 27605
Description: A version of eqrelrdv2 5111 with explicit non-free declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Hypotheses
Ref Expression
eqrelrd2.1  |-  F/ x ph
eqrelrd2.2  |-  F/ y
ph
eqrelrd2.3  |-  F/_ x A
eqrelrd2.4  |-  F/_ y A
eqrelrd2.5  |-  F/_ x B
eqrelrd2.6  |-  F/_ y B
eqrelrd2.7  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
Assertion
Ref Expression
eqrelrd2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem eqrelrd2
StepHypRef Expression
1 eqrelrd2.1 . . . 4  |-  F/ x ph
2 eqrelrd2.2 . . . . 5  |-  F/ y
ph
3 eqrelrd2.7 . . . . 5  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
42, 3alrimi 1878 . . . 4  |-  ( ph  ->  A. y ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B ) )
51, 4alrimi 1878 . . 3  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
65adantl 466 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
7 eqrelrd2.3 . . . . . 6  |-  F/_ x A
8 eqrelrd2.4 . . . . . 6  |-  F/_ y A
9 eqrelrd2.5 . . . . . 6  |-  F/_ x B
10 eqrelrd2.6 . . . . . 6  |-  F/_ y B
111, 2, 7, 8, 9, 10ssrelf 27604 . . . . 5  |-  ( Rel 
A  ->  ( A  C_  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
) ) )
121, 2, 9, 10, 7, 8ssrelf 27604 . . . . 5  |-  ( Rel 
B  ->  ( B  C_  A  <->  A. x A. y
( <. x ,  y
>.  e.  B  ->  <. x ,  y >.  e.  A
) ) )
1311, 12bi2anan9 873 . . . 4  |-  ( ( Rel  A  /\  Rel  B )  ->  ( ( A  C_  B  /\  B  C_  A )  <->  ( A. x A. y ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
)  /\  A. x A. y ( <. x ,  y >.  e.  B  -> 
<. x ,  y >.  e.  A ) ) ) )
14 eqss 3514 . . . 4  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
15 2albiim 1701 . . . 4  |-  ( A. x A. y ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )  <->  ( A. x A. y ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
)  /\  A. x A. y ( <. x ,  y >.  e.  B  -> 
<. x ,  y >.  e.  A ) ) )
1613, 14, 153bitr4g 288 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
1716adantr 465 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
186, 17mpbird 232 1  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393    = wceq 1395   F/wnf 1617    e. wcel 1819   F/_wnfc 2605    C_ wss 3471   <.cop 4038   Rel wrel 5013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516  df-xp 5014  df-rel 5015
This theorem is referenced by:  fpwrelmap  27706
  Copyright terms: Public domain W3C validator