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Theorem fliftrel 6189
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 4855 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( R  X.  S
) )
52, 3, 4syl2anc 659 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  ( R  X.  S ) )
6 eqid 2402 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
75, 6fmptd 6033 . . 3  |-  ( ph  ->  ( x  e.  X  |-> 
<. A ,  B >. ) : X --> ( R  X.  S ) )
8 frn 5720 . . 3  |-  ( ( x  e.  X  |->  <. A ,  B >. ) : X --> ( R  X.  S )  ->  ran  ( x  e.  X  |-> 
<. A ,  B >. ) 
C_  ( R  X.  S ) )
97, 8syl 17 . 2  |-  ( ph  ->  ran  ( x  e.  X  |->  <. A ,  B >. )  C_  ( R  X.  S ) )
101, 9syl5eqss 3486 1  |-  ( ph  ->  F  C_  ( R  X.  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3414   <.cop 3978    |-> cmpt 4453    X. cxp 4821   ran crn 4824   -->wf 5565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577
This theorem is referenced by:  fliftcnv  6192  fliftfun  6193  fliftf  6196  qliftrel  7430  fmucndlem  21086  pi1xfrcnv  21849
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