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Theorem fliftrel 5999
Description:  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
flift.2  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
flift.3  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
Assertion
Ref Expression
fliftrel  |-  ( ph  ->  F  C_  ( R  X.  S ) )
Distinct variable groups:    x, R    ph, x    x, X    x, S
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem fliftrel
StepHypRef Expression
1 flift.1 . 2  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
2 flift.2 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
3 flift.3 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
4 opelxpi 4869 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( R  X.  S
) )
52, 3, 4syl2anc 661 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  ( R  X.  S ) )
6 eqid 2441 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
75, 6fmptd 5865 . . 3  |-  ( ph  ->  ( x  e.  X  |-> 
<. A ,  B >. ) : X --> ( R  X.  S ) )
8 frn 5563 . . 3  |-  ( ( x  e.  X  |->  <. A ,  B >. ) : X --> ( R  X.  S )  ->  ran  ( x  e.  X  |-> 
<. A ,  B >. ) 
C_  ( R  X.  S ) )
97, 8syl 16 . 2  |-  ( ph  ->  ran  ( x  e.  X  |->  <. A ,  B >. )  C_  ( R  X.  S ) )
101, 9syl5eqss 3398 1  |-  ( ph  ->  F  C_  ( R  X.  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3326   <.cop 3881    e. cmpt 4348    X. cxp 4836   ran crn 4839   -->wf 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424
This theorem is referenced by:  fliftcnv  6002  fliftfun  6003  fliftf  6006  qliftrel  7180  fmucndlem  19864  pi1xfrcnv  20627
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