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Theorem fliftrel 6022
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 4902 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( R  X.  S
) )
52, 3, 4syl2anc 643 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  ( R  X.  S ) )
6 eqid 2435 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
75, 6fmptd 5885 . . 3  |-  ( ph  ->  ( x  e.  X  |-> 
<. A ,  B >. ) : X --> ( R  X.  S ) )
8 frn 5589 . . 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 3384 1  |-  ( ph  ->  F  C_  ( R  X.  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   <.cop 3809    e. cmpt 4258    X. cxp 4868   ran crn 4871   -->wf 5442
This theorem is referenced by:  fliftcnv  6025  fliftfun  6026  fliftf  6029  qliftrel  6978  fmucndlem  18311  pi1xfrcnv  19072
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454
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