Theorem suplocexprlemml 8073

Description: Lemma for suplocexpr 8082. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	|- ( ph -> E. x x e. A )
suplocexpr.ub	|- ( ph -> E. x e. P. A. y e. A y <P x )
suplocexpr.loc	|- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )

Assertion

Ref	Expression
suplocexprlemml	|- ( ph -> E. s e. Q. s e. U. ( 1st " A ) )

Distinct variable groups:   A, s, x, y   ph, s, x, y

Allowed substitution hints:   ph( z)   A( z)

Proof of Theorem suplocexprlemml

Step	Hyp	Ref	Expression
1		suplocexpr.m	. . 3 |- ( ph -> E. x x e. A )
2		suplocexpr.ub	. . . . . . 7 |- ( ph -> E. x e. P. A. y e. A y <P x )
3		suplocexpr.loc	. . . . . . 7 |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )
4	1, 2, 3	suplocexprlemss 8072	. . . . . 6 |- ( ph -> A C_ P. )
5	4	sselda 3248	. . . . 5 |- ( ( ph /\ x e. A ) -> x e. P. )
6		prop 7832	. . . . 5 |- ( x e. P. -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. )
7		prml 7834	. . . . 5 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. -> E. s e. Q. s e. ( 1st ` x ) )
8	5, 6, 7	3syl 17	. . . 4 |- ( ( ph /\ x e. A ) -> E. s e. Q. s e. ( 1st ` x ) )
9	8	ralrimiva 2623	. . 3 |- ( ph -> A. x e. A E. s e. Q. s e. ( 1st ` x ) )
10		r19.2m 3611	. . 3 |- ( ( E. x x e. A /\ A. x e. A E. s e. Q. s e. ( 1st ` x ) ) -> E. x e. A E. s e. Q. s e. ( 1st ` x ) )
11	1, 9, 10	syl2anc 415	. 2 |- ( ph -> E. x e. A E. s e. Q. s e. ( 1st ` x ) )
12		suplocexprlemell 8070	. . . 4 |- ( s e. U. ( 1st " A ) <-> E. x e. A s e. ( 1st ` x ) )
13	12	rexbii 2557	. . 3 |- ( E. s e. Q. s e. U. ( 1st " A ) <-> E. s e. Q. E. x e. A s e. ( 1st ` x ) )
14		rexcom 2715	. . 3 |- ( E. s e. Q. E. x e. A s e. ( 1st ` x ) <-> E. x e. A E. s e. Q. s e. ( 1st ` x ) )
15	13, 14	bitri 184	. 2 |- ( E. s e. Q. s e. U. ( 1st " A ) <-> E. x e. A E. s e. Q. s e. ( 1st ` x ) )
16	11, 15	sylibr 134	1 |- ( ph -> E. s e. Q. s e. U. ( 1st " A ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   <.cop 3708   U.cuni 3930   class class class wbr 4125   "cima 4772   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   P.cnp 7648   <P cltp 7652

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-in1 623  ax-in2 624  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-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem 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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-qs 6803  df-ni 7661  df-nqqs 7705  df-inp 7823  df-iltp 7827

This theorem is referenced by:  suplocexprlemex  8079