Theorem suplocexprlemell 7923

Description: Lemma for suplocexpr 7935. 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 6315	. . . . 5 |- 1st : _V -onto-> _V
2		fofn 5558	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . 4 |- 1st Fn _V
4		ssv 3247	. . . 4 |- A C_ _V
5		fnssres 5442	. . . 4 |- ( ( 1st Fn _V /\ A C_ _V ) -> ( 1st |` A ) Fn A )
6	3, 4, 5	mp2an 426	. . 3 |- ( 1st |` A ) Fn A
7		eluniimadm 5901	. . 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 5044	. . . 4 |- ( ( 1st |` A ) " A ) = ( 1st " A )
10	9	unieqi 3901	. . 3 |- U. ( ( 1st |` A ) " A ) = U. ( 1st " A )
11	10	eleq2i 2296	. 2 |- ( B e. U. ( ( 1st |` A ) " A ) <-> B e. U. ( 1st " A ) )
12		fvres 5659	. . . 4 |- ( x e. A -> ( ( 1st |` A ) ` x ) = ( 1st ` x ) )
13	12	eleq2d 2299	. . 3 |- ( x e. A -> ( B e. ( ( 1st |` A ) ` x ) <-> B e. ( 1st ` x ) ) )
14	13	rexbiia 2545	. 2 |- ( E. x e. A B e. ( ( 1st |` A ) ` x ) <-> E. x e. A B e. ( 1st ` x ) )
15	8, 11, 14	3bitr3i 210	1 |- ( B e. U. ( 1st " A ) <-> E. x e. A B e. ( 1st ` x ) )

Syntax hints:   <-> wb 105   e. wcel 2200   E.wrex 2509   _Vcvv 2800   C_ wss 3198   U.cuni 3891   |` cres 4725   "cima 4726   Fn wfn 5319   -onto->wfo 5322   ` cfv 5324   1stc1st 6296

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-1st 6298

This theorem is referenced by:  suplocexprlemml  7926  suplocexprlemrl  7927  suplocexprlemdisj  7930  suplocexprlemloc  7931  suplocexprlemex  7932  suplocexprlemlub  7934