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Theorem suplocexprlemell 7703
Description: Lemma for suplocexpr 7715. 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 6152 . . . . 5  |-  1st : _V -onto-> _V
2 fofn 5436 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . . 4  |-  1st  Fn  _V
4 ssv 3177 . . . 4  |-  A  C_  _V
5 fnssres 5325 . . . 4  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( 1st  |`  A )  Fn  A )
63, 4, 5mp2an 426 . . 3  |-  ( 1st  |`  A )  Fn  A
7 eluniimadm 5760 . . 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 4936 . . . 4  |-  ( ( 1st  |`  A ) " A )  =  ( 1st " A )
109unieqi 3817 . . 3  |-  U. (
( 1st  |`  A )
" A )  = 
U. ( 1st " A
)
1110eleq2i 2244 . 2  |-  ( B  e.  U. ( ( 1st  |`  A ) " A )  <->  B  e.  U. ( 1st " A
) )
12 fvres 5535 . . . 4  |-  ( x  e.  A  ->  (
( 1st  |`  A ) `
 x )  =  ( 1st `  x
) )
1312eleq2d 2247 . . 3  |-  ( x  e.  A  ->  ( B  e.  ( ( 1st  |`  A ) `  x )  <->  B  e.  ( 1st `  x ) ) )
1413rexbiia 2492 . 2  |-  ( E. x  e.  A  B  e.  ( ( 1st  |`  A ) `
 x )  <->  E. x  e.  A  B  e.  ( 1st `  x ) )
158, 11, 143bitr3i 210 1  |-  ( B  e.  U. ( 1st " A )  <->  E. x  e.  A  B  e.  ( 1st `  x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    e. wcel 2148   E.wrex 2456   _Vcvv 2737    C_ wss 3129   U.cuni 3807    |` cres 4625   "cima 4626    Fn wfn 5207   -onto->wfo 5210   ` cfv 5212   1stc1st 6133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fo 5218  df-fv 5220  df-1st 6135
This theorem is referenced by:  suplocexprlemml  7706  suplocexprlemrl  7707  suplocexprlemdisj  7710  suplocexprlemloc  7711  suplocexprlemex  7712  suplocexprlemlub  7714
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