MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relmpt2opab Structured version   Unicode version

Theorem relmpt2opab 6755
Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
relmpt2opab.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  { <. z ,  w >.  |  ph } )
Assertion
Ref Expression
relmpt2opab  |-  Rel  ( C F D )
Distinct variable groups:    x, w, y, z    y, B    x, A, y
Allowed substitution hints:    ph( x, y, z, w)    A( z, w)    B( x, z, w)    C( x, y, z, w)    D( x, y, z, w)    F( x, y, z, w)

Proof of Theorem relmpt2opab
StepHypRef Expression
1 relopab 5064 . . . . 5  |-  Rel  { <. z ,  w >.  | 
ph }
2 df-rel 4945 . . . . 5  |-  ( Rel 
{ <. z ,  w >.  |  ph }  <->  { <. z ,  w >.  |  ph }  C_  ( _V  X.  _V ) )
31, 2mpbi 208 . . . 4  |-  { <. z ,  w >.  |  ph }  C_  ( _V  X.  _V )
43rgen2w 2892 . . 3  |-  A. x  e.  A  A. y  e.  B  { <. z ,  w >.  |  ph }  C_  ( _V  X.  _V )
5 relmpt2opab.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  { <. z ,  w >.  |  ph } )
65ovmptss 6754 . . 3  |-  ( A. x  e.  A  A. y  e.  B  { <. z ,  w >.  | 
ph }  C_  ( _V  X.  _V )  -> 
( C F D )  C_  ( _V  X.  _V ) )
74, 6ax-mp 5 . 2  |-  ( C F D )  C_  ( _V  X.  _V )
8 df-rel 4945 . 2  |-  ( Rel  ( C F D )  <->  ( C F D )  C_  ( _V  X.  _V ) )
97, 8mpbir 209 1  |-  Rel  ( C F D )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   A.wral 2795   _Vcvv 3068    C_ wss 3426   {copab 4447    X. cxp 4936   Rel wrel 4943  (class class class)co 6190    |-> cmpt2 6192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678
This theorem is referenced by:  brovmpt2ex  6841  relfunc  14874  releqg  15830  releupa  23720  relwlk  30419
  Copyright terms: Public domain W3C validator