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Theorem fliftrel 5962
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 4843 . . . . 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 2380 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
75, 6fmptd 5825 . . 3  |-  ( ph  ->  ( x  e.  X  |-> 
<. A ,  B >. ) : X --> ( R  X.  S ) )
8 frn 5530 . . 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 3328 1  |-  ( ph  ->  F  C_  ( R  X.  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3256   <.cop 3753    e. cmpt 4200    X. cxp 4809   ran crn 4812   -->wf 5383
This theorem is referenced by:  fliftcnv  5965  fliftfun  5966  fliftf  5969  qliftrel  6915  fmucndlem  18235  pi1xfrcnv  18946
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395
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