Theorem suplocexprlemell 8086

Description: Lemma for suplocexpr 8098. 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 6395	. . . . 5 |- 1st : _V -onto-> _V
2		fofn 5620	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . 4 |- 1st Fn _V
4		ssv 3270	. . . 4 |- A C_ _V
5		fnssres 5499	. . . 4 |- ( ( 1st Fn _V /\ A C_ _V ) -> ( 1st |` A ) Fn A )
6	3, 4, 5	mp2an 430	. . 3 |- ( 1st |` A ) Fn A
7		eluniimadm 5974	. . 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 5098	. . . 4 |- ( ( 1st |` A ) " A ) = ( 1st " A )
10	9	unieqi 3945	. . 3 |- U. ( ( 1st |` A ) " A ) = U. ( 1st " A )
11	10	eleq2i 2305	. 2 |- ( B e. U. ( ( 1st |` A ) " A ) <-> B e. U. ( 1st " A ) )
12		fvres 5722	. . . 4 |- ( x e. A -> ( ( 1st |` A ) ` x ) = ( 1st ` x ) )
13	12	eleq2d 2308	. . 3 |- ( x e. A -> ( B e. ( ( 1st |` A ) ` x ) <-> B e. ( 1st ` x ) ) )
14	13	rexbiia 2565	. 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 ) )

This proof depends on syntax axioms:   <-> wb 105   e. wcel 2209   E.wrex 2529   _Vcvv 2821   C_ wss 3220   U.cuni 3935   |` cres 4778   "cima 4779   Fn wfn 5375   -onto->wfo 5378   ` cfv 5380   1stc1st 6376

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fo 5386  df-fv 5388  df-1st 6378

This theorem is used by:  suplocexprlemml  8089  suplocexprlemrl  8090  suplocexprlemdisj  8093  suplocexprlemloc  8094  suplocexprlemex  8095  suplocexprlemlub  8097