Theorem suplocexprlemell 7654

Description: Lemma for suplocexpr 7666. 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 6125	. . . . 5 |- 1st : _V -onto-> _V
2		fofn 5412	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . 4 |- 1st Fn _V
4		ssv 3164	. . . 4 |- A C_ _V
5		fnssres 5301	. . . 4 |- ( ( 1st Fn _V /\ A C_ _V ) -> ( 1st |` A ) Fn A )
6	3, 4, 5	mp2an 423	. . 3 |- ( 1st |` A ) Fn A
7		eluniimadm 5733	. . 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 4917	. . . 4 |- ( ( 1st |` A ) " A ) = ( 1st " A )
10	9	unieqi 3799	. . 3 |- U. ( ( 1st |` A ) " A ) = U. ( 1st " A )
11	10	eleq2i 2233	. 2 |- ( B e. U. ( ( 1st |` A ) " A ) <-> B e. U. ( 1st " A ) )
12		fvres 5510	. . . 4 |- ( x e. A -> ( ( 1st |` A ) ` x ) = ( 1st ` x ) )
13	12	eleq2d 2236	. . 3 |- ( x e. A -> ( B e. ( ( 1st |` A ) ` x ) <-> B e. ( 1st ` x ) ) )
14	13	rexbiia 2481	. 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 2136   E.wrex 2445   _Vcvv 2726   C_ wss 3116   U.cuni 3789   |` cres 4606   "cima 4607   Fn wfn 5183   -onto->wfo 5186   ` cfv 5188   1stc1st 6106

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-1st 6108

This theorem is referenced by:  suplocexprlemml  7657  suplocexprlemrl  7658  suplocexprlemdisj  7661  suplocexprlemloc  7662  suplocexprlemex  7663  suplocexprlemlub  7665