Theorem suplocexprlemell 7545

Description: Lemma for suplocexpr 7557. 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

Step	Hyp	Ref	Expression
1		fo1st 6063	. . . . 5 |- 1st : _V -onto-> _V
2		fofn 5355	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . 4 |- 1st Fn _V
4		ssv 3124	. . . 4 |- A C_ _V
5		fnssres 5244	. . . 4 |- ( ( 1st Fn _V /\ A C_ _V ) -> ( 1st |` A ) Fn A )
6	3, 4, 5	mp2an 423	. . 3 |- ( 1st |` A ) Fn A
7		eluniimadm 5674	. . 3 |- ( ( 1st |` A ) Fn A -> ( B e. U. ( ( 1st |` A ) " A ) <-> E. x e. A B e. ( ( 1st |` A ) ` x ) ) )
8	6, 7	ax-mp 5	. 2 |- ( B e. U. ( ( 1st |` A ) " A ) <-> E. x e. A B e. ( ( 1st |` A ) ` x ) )
9		resima 4860	. . . 4 |- ( ( 1st |` A ) " A ) = ( 1st " A )
10	9	unieqi 3754	. . 3 |- U. ( ( 1st |` A ) " A ) = U. ( 1st " A )
11	10	eleq2i 2207	. 2 |- ( B e. U. ( ( 1st |` A ) " A ) <-> B e. U. ( 1st " A ) )
12		fvres 5453	. . . 4 |- ( x e. A -> ( ( 1st |` A ) ` x ) = ( 1st ` x ) )
13	12	eleq2d 2210	. . 3 |- ( x e. A -> ( B e. ( ( 1st |` A ) ` x ) <-> B e. ( 1st ` x ) ) )
14	13	rexbiia 2453	. 2 |- ( E. x e. A B e. ( ( 1st |` A ) ` x ) <-> E. x e. A B e. ( 1st ` x ) )
15	8, 11, 14	3bitr3i 209	1 |- ( B e. U. ( 1st " A ) <-> E. x e. A B e. ( 1st ` x ) )

Syntax hints:   <-> wb 104   e. wcel 1481   E.wrex 2418   _Vcvv 2689   C_ wss 3076   U.cuni 3744   |` cres 4549   "cima 4550   Fn wfn 5126   -onto->wfo 5129   ` cfv 5131   1stc1st 6044

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139  df-1st 6046

This theorem is referenced by:  suplocexprlemml  7548  suplocexprlemrl  7549  suplocexprlemdisj  7552  suplocexprlemloc  7553  suplocexprlemex  7554  suplocexprlemlub  7556