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Theorem suplocexprlemell 7514
Description: Lemma for suplocexpr 7526. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
Assertion
Ref Expression
suplocexprlemell  |-  ( B  e.  U. ( 1st " A )  <->  E. x  e.  A  B  e.  ( 1st `  x ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem suplocexprlemell
StepHypRef Expression
1 fo1st 6048 . . . . 5  |-  1st : _V -onto-> _V
2 fofn 5342 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . . 4  |-  1st  Fn  _V
4 ssv 3114 . . . 4  |-  A  C_  _V
5 fnssres 5231 . . . 4  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( 1st  |`  A )  Fn  A )
63, 4, 5mp2an 422 . . 3  |-  ( 1st  |`  A )  Fn  A
7 eluniimadm 5659 . . 3  |-  ( ( 1st  |`  A )  Fn  A  ->  ( B  e.  U. ( ( 1st  |`  A ) " A )  <->  E. x  e.  A  B  e.  ( ( 1st  |`  A ) `
 x ) ) )
86, 7ax-mp 5 . 2  |-  ( B  e.  U. ( ( 1st  |`  A ) " A )  <->  E. x  e.  A  B  e.  ( ( 1st  |`  A ) `
 x ) )
9 resima 4847 . . . 4  |-  ( ( 1st  |`  A ) " A )  =  ( 1st " A )
109unieqi 3741 . . 3  |-  U. (
( 1st  |`  A )
" A )  = 
U. ( 1st " A
)
1110eleq2i 2204 . 2  |-  ( B  e.  U. ( ( 1st  |`  A ) " A )  <->  B  e.  U. ( 1st " A
) )
12 fvres 5438 . . . 4  |-  ( x  e.  A  ->  (
( 1st  |`  A ) `
 x )  =  ( 1st `  x
) )
1312eleq2d 2207 . . 3  |-  ( x  e.  A  ->  ( B  e.  ( ( 1st  |`  A ) `  x )  <->  B  e.  ( 1st `  x ) ) )
1413rexbiia 2448 . 2  |-  ( E. x  e.  A  B  e.  ( ( 1st  |`  A ) `
 x )  <->  E. x  e.  A  B  e.  ( 1st `  x ) )
158, 11, 143bitr3i 209 1  |-  ( B  e.  U. ( 1st " A )  <->  E. x  e.  A  B  e.  ( 1st `  x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1480   E.wrex 2415   _Vcvv 2681    C_ wss 3066   U.cuni 3731    |` cres 4536   "cima 4537    Fn wfn 5113   -onto->wfo 5116   ` cfv 5118   1stc1st 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-1st 6031
This theorem is referenced by:  suplocexprlemml  7517  suplocexprlemrl  7518  suplocexprlemdisj  7521  suplocexprlemloc  7522  suplocexprlemex  7523  suplocexprlemlub  7525
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