Theorem suplocexprlemell 7856

Description: Lemma for suplocexpr 7868. 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 6261	. . . . 5 |- 1st : _V -onto-> _V
2		fofn 5517	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . . 4 |- 1st Fn _V
4		ssv 3219	. . . 4 |- A C_ _V
5		fnssres 5403	. . . 4 |- ( ( 1st Fn _V /\ A C_ _V ) -> ( 1st |` A ) Fn A )
6	3, 4, 5	mp2an 426	. . 3 |- ( 1st |` A ) Fn A
7		eluniimadm 5852	. . 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 5006	. . . 4 |- ( ( 1st |` A ) " A ) = ( 1st " A )
10	9	unieqi 3869	. . 3 |- U. ( ( 1st |` A ) " A ) = U. ( 1st " A )
11	10	eleq2i 2273	. 2 |- ( B e. U. ( ( 1st |` A ) " A ) <-> B e. U. ( 1st " A ) )
12		fvres 5618	. . . 4 |- ( x e. A -> ( ( 1st |` A ) ` x ) = ( 1st ` x ) )
13	12	eleq2d 2276	. . 3 |- ( x e. A -> ( B e. ( ( 1st |` A ) ` x ) <-> B e. ( 1st ` x ) ) )
14	13	rexbiia 2522	. 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 2177   E.wrex 2486   _Vcvv 2773   C_ wss 3170   U.cuni 3859   |` cres 4690   "cima 4691   Fn wfn 5280   -onto->wfo 5283   ` cfv 5285   1stc1st 6242

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-fo 5291  df-fv 5293  df-1st 6244

This theorem is referenced by:  suplocexprlemml  7859  suplocexprlemrl  7860  suplocexprlemdisj  7863  suplocexprlemloc  7864  suplocexprlemex  7865  suplocexprlemlub  7867