Theorem frminex 15773

Description: If an element of a founded set satisfies a property ph, then there is a minimal element that satisfies ph.

Hypotheses

Ref	Expression
frminex.1	|- A e. _V
frminex.2	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
frminex	|- (R Fr A -> (E.x e. A ph -> E.x e. A (ph /\ A.y e. A (ps -> -. yRx))))

Distinct variable groups:   x,A,y   x,R,y   ph,y   ps,x

Proof of Theorem frminex

Step	Hyp	Ref	Expression
1		frminex.1	. . . . 5 |- A e. _V
2	1	rabex 3461	. . . 4 |- {x e. A | ph} e. _V
3		ssrab2 2692	. . . 4 |- {x e. A | ph} C_ A
4		fri 3626	. . . . . 6 |- ((({x e. A | ph} e. _V /\ R Fr A) /\ ({x e. A | ph} C_ A /\ {x e. A | ph} =/= (/))) -> E.z e. {x e. A | ph}A.y e. {x e. A | ph} -. yRz)
5		ax-17 1317	. . . . . . . 8 |- (y e. {x e. A | ph} -> A.z y e. {x e. A | ph})
6		hbrab1 2257	. . . . . . . 8 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
7		ax-17 1317	. . . . . . . . 9 |- (-. yRz -> A.x -. yRz)
8	6, 7	hbral 2146	. . . . . . . 8 |- (A.y e. {x e. A | ph} -. yRz -> A.xA.y e. {x e. A | ph} -. yRz)
9		ax-17 1317	. . . . . . . 8 |- (A.y e. {x e. A | ph} -. yRx -> A.zA.y e. {x e. A | ph} -. yRx)
10		breq2 3342	. . . . . . . . . 10 |- (z = x -> (yRz <-> yRx))
11	10	notbid 673	. . . . . . . . 9 |- (z = x -> (-. yRz <-> -. yRx))
12	11	ralbidv 2123	. . . . . . . 8 |- (z = x -> (A.y e. {x e. A | ph} -. yRz <-> A.y e. {x e. A | ph} -. yRx))
13	5, 6, 8, 9, 12	cbvrexf 2277	. . . . . . 7 |- (E.z e. {x e. A | ph}A.y e. {x e. A | ph} -. yRz <-> E.x e. {x e. A | ph}A.y e. {x e. A | ph} -. yRx)
14		frminex.2	. . . . . . . . . . . . . . 15 |- (x = y -> (ph <-> ps))
15	14	elrab 2414	. . . . . . . . . . . . . 14 |- (y e. {x e. A | ph} <-> (y e. A /\ ps))
16	15	imbi1i 203	. . . . . . . . . . . . 13 |- ((y e. {x e. A | ph} -> -. yRx) <-> ((y e. A /\ ps) -> -. yRx))
17		pm3.3 375	. . . . . . . . . . . . 13 |- (((y e. A /\ ps) -> -. yRx) -> (y e. A -> (ps -> -. yRx)))
18	16, 17	sylbi 216	. . . . . . . . . . . 12 |- ((y e. {x e. A | ph} -> -. yRx) -> (y e. A -> (ps -> -. yRx)))
19	18	ralimi2 2165	. . . . . . . . . . 11 |- (A.y e. {x e. A | ph} -. yRx -> A.y e. A (ps -> -. yRx))
20	19	anim2i 362	. . . . . . . . . 10 |- (((x e. A /\ ph) /\ A.y e. {x e. A | ph} -. yRx) -> ((x e. A /\ ph) /\ A.y e. A (ps -> -. yRx)))
21		rabid 2253	. . . . . . . . . 10 |- (x e. {x e. A | ph} <-> (x e. A /\ ph))
22	20, 21	sylanb 498	. . . . . . . . 9 |- ((x e. {x e. A | ph} /\ A.y e. {x e. A | ph} -. yRx) -> ((x e. A /\ ph) /\ A.y e. A (ps -> -. yRx)))
23		anass 487	. . . . . . . . 9 |- (((x e. A /\ ph) /\ A.y e. A (ps -> -. yRx)) <-> (x e. A /\ (ph /\ A.y e. A (ps -> -. yRx))))
24	22, 23	sylib 215	. . . . . . . 8 |- ((x e. {x e. A | ph} /\ A.y e. {x e. A | ph} -. yRx) -> (x e. A /\ (ph /\ A.y e. A (ps -> -. yRx))))
25	24	reximi2 2197	. . . . . . 7 |- (E.x e. {x e. A | ph}A.y e. {x e. A | ph} -. yRx -> E.x e. A (ph /\ A.y e. A (ps -> -. yRx)))
26	13, 25	sylbi 216	. . . . . 6 |- (E.z e. {x e. A | ph}A.y e. {x e. A | ph} -. yRz -> E.x e. A (ph /\ A.y e. A (ps -> -. yRx)))
27	4, 26	syl 12	. . . . 5 |- ((({x e. A | ph} e. _V /\ R Fr A) /\ ({x e. A | ph} C_ A /\ {x e. A | ph} =/= (/))) -> E.x e. A (ph /\ A.y e. A (ps -> -. yRx)))
28	27	an4s 566	. . . 4 |- ((({x e. A | ph} e. _V /\ {x e. A | ph} C_ A) /\ (R Fr A /\ {x e. A | ph} =/= (/))) -> E.x e. A (ph /\ A.y e. A (ps -> -. yRx)))
29	2, 3, 28	mpanl12 773	. . 3 |- ((R Fr A /\ {x e. A | ph} =/= (/)) -> E.x e. A (ph /\ A.y e. A (ps -> -. yRx)))
30	29	ex 402	. 2 |- (R Fr A -> ({x e. A | ph} =/= (/) -> E.x e. A (ph /\ A.y e. A (ps -> -. yRx))))
31		rabn0 2893	. 2 |- ({x e. A | ph} =/= (/) <-> E.x e. A ph)
32	30, 31	syl5ibr 224	1 |- (R Fr A -> (E.x e. A ph -> E.x e. A (ph /\ A.y e. A (ps -> -. yRx))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  (/)c0 2875   class class class wbr 3338   Fr wfr 3623

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-fr 3625

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