Theorem suplocexprlemml 7859

Description: Lemma for suplocexpr 7868. 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 7858	. . . . . 6 |- ( ph -> A C_ P. )
5	4	sselda 3197	. . . . 5 |- ( ( ph /\ x e. A ) -> x e. P. )
6		prop 7618	. . . . 5 |- ( x e. P. -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. )
7		prml 7620	. . . . 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 2580	. . 3 |- ( ph -> A. x e. A E. s e. Q. s e. ( 1st ` x ) )
10		r19.2m 3551	. . 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 411	. 2 |- ( ph -> E. x e. A E. s e. Q. s e. ( 1st ` x ) )
12		suplocexprlemell 7856	. . . 4 |- ( s e. U. ( 1st " A ) <-> E. x e. A s e. ( 1st ` x ) )
13	12	rexbii 2514	. . 3 |- ( E. s e. Q. s e. U. ( 1st " A ) <-> E. s e. Q. E. x e. A s e. ( 1st ` x ) )
14		rexcom 2671	. . 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 710   E.wex 1516   e. wcel 2177   A.wral 2485   E.wrex 2486   <.cop 3641   U.cuni 3859   class class class wbr 4054   "cima 4691   ` cfv 5285   1stc1st 6242   2ndc2nd 6243   Q.cnq 7423   P.cnp 7434   <P cltp 7438

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 615  ax-in2 616  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-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-iinf 4649

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-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  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-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-iom 4652  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-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-1st 6244  df-2nd 6245  df-qs 6644  df-ni 7447  df-nqqs 7491  df-inp 7609  df-iltp 7613

This theorem is referenced by:  suplocexprlemex  7865