Theorem acdc2lem1 7580

Description: Lemma for acdc2 7582.

Hypotheses

Ref	Expression
acdc2lem.1	|- A e. V
acdc2lem.2	|- S = {<.<.x, y>., z>. | ((x e. A /\ y e. NN) /\ z = U.{v e. (yFx) | A.u e. (yFx) -. urv})}
acdc2lem.3	|- G = (S seq1 ({<.1, c>.} u. (I |` (NN \ {1}))))

Assertion

Ref	Expression
acdc2lem1	|- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> ((XSK) e. (KFX) /\ (XSK) e. A))

Distinct variable groups:   v,u,x,y,z,A   u,F,v,x,y,z   u,G,v,x,y,z   x,c,y,z   u,r,v,x,y,z   u,K,v,x,y,z   u,X,v,x,y,z

Proof of Theorem acdc2lem1

Step	Hyp	Ref	Expression
1		oprex 4041	. . . . . . 7 |- (KFX) e. V
2	1	rabex 2780	. . . . . 6 |- {v e. (KFX) | A.u e. (KFX) -. urv} e. V
3	2	uniex 2926	. . . . 5 |- U.{v e. (KFX) | A.u e. (KFX) -. urv} e. V
4		opreq2 4027	. . . . . . 7 |- (x = X -> (yFx) = (yFX))
5		rabeq 1856	. . . . . . . 8 |- ((yFx) = (yFX) -> {v e. (yFx) | A.u e. (yFx) -. urv} = {v e. (yFX) | A.u e. (yFx) -. urv})
6		raleq1 1833	. . . . . . . . 9 |- ((yFx) = (yFX) -> (A.u e. (yFx) -. urv <-> A.u e. (yFX) -. urv))
7	6	rabbisdv 1854	. . . . . . . 8 |- ((yFx) = (yFX) -> {v e. (yFX) | A.u e. (yFx) -. urv} = {v e. (yFX) | A.u e. (yFX) -. urv})
8	5, 7	eqtrd 1554	. . . . . . 7 |- ((yFx) = (yFX) -> {v e. (yFx) | A.u e. (yFx) -. urv} = {v e. (yFX) | A.u e. (yFX) -. urv})
9	4, 8	syl 10	. . . . . 6 |- (x = X -> {v e. (yFx) | A.u e. (yFx) -. urv} = {v e. (yFX) | A.u e. (yFX) -. urv})
10	9	unieqd 2566	. . . . 5 |- (x = X -> U.{v e. (yFx) | A.u e. (yFx) -. urv} = U.{v e. (yFX) | A.u e. (yFX) -. urv})
11		opreq1 4026	. . . . . . 7 |- (y = K -> (yFX) = (KFX))
12		rabeq 1856	. . . . . . . 8 |- ((yFX) = (KFX) -> {v e. (yFX) | A.u e. (yFX) -. urv} = {v e. (KFX) | A.u e. (yFX) -. urv})
13		raleq1 1833	. . . . . . . . 9 |- ((yFX) = (KFX) -> (A.u e. (yFX) -. urv <-> A.u e. (KFX) -. urv))
14	13	rabbisdv 1854	. . . . . . . 8 |- ((yFX) = (KFX) -> {v e. (KFX) | A.u e. (yFX) -. urv} = {v e. (KFX) | A.u e. (KFX) -. urv})
15	12, 14	eqtrd 1554	. . . . . . 7 |- ((yFX) = (KFX) -> {v e. (yFX) | A.u e. (yFX) -. urv} = {v e. (KFX) | A.u e. (KFX) -. urv})
16	11, 15	syl 10	. . . . . 6 |- (y = K -> {v e. (yFX) | A.u e. (yFX) -. urv} = {v e. (KFX) | A.u e. (KFX) -. urv})
17	16	unieqd 2566	. . . . 5 |- (y = K -> U.{v e. (yFX) | A.u e. (yFX) -. urv} = U.{v e. (KFX) | A.u e. (KFX) -. urv})
18		acdc2lem.2	. . . . 5 |- S = {<.<.x, y>., z>. | ((x e. A /\ y e. NN) /\ z = U.{v e. (yFx) | A.u e. (yFx) -. urv})}
19	3, 10, 17, 18	oprabval2 4086	. . . 4 |- ((X e. A /\ K e. NN) -> (XSK) = U.{v e. (KFX) | A.u e. (KFX) -. urv})
20	19	adantl 397	. . 3 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> (XSK) = U.{v e. (KFX) | A.u e. (KFX) -. urv})
21	1	wereucl 3003	. . . 4 |- ((r We A /\ (KFX) (_ A /\ (KFX) =/= (/)) -> U.{v e. (KFX) | A.u e. (KFX) -. urv} e. (KFX))
22		simpll 421	. . . 4 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> r We A)
23		foprrn 4093	. . . . . . . . 9 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ K e. NN /\ X e. A) -> (KFX) e. (P~A \ {(/)}))
24	23	3com23 851	. . . . . . . 8 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ X e. A /\ K e. NN) -> (KFX) e. (P~A \ {(/)}))
25	24	3expb 846	. . . . . . 7 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ (X e. A /\ K e. NN)) -> (KFX) e. (P~A \ {(/)}))
26		eldifi 2213	. . . . . . 7 |- ((KFX) e. (P~A \ {(/)}) -> (KFX) e. P~A)
27	25, 26	syl 10	. . . . . 6 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ (X e. A /\ K e. NN)) -> (KFX) e. P~A)
28		elpwi 2458	. . . . . 6 |- ((KFX) e. P~A -> (KFX) (_ A)
29	27, 28	syl 10	. . . . 5 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ (X e. A /\ K e. NN)) -> (KFX) (_ A)
30	29	adantll 401	. . . 4 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> (KFX) (_ A)
31		eldifn 2214	. . . . . . 7 |- ((KFX) e. (P~A \ {(/)}) -> -. (KFX) e. {(/)})
32		id 59	. . . . . . . . 9 |- ((KFX) = (/) -> (KFX) = (/))
33		0ex 2766	. . . . . . . . . 10 |- (/) e. V
34	33	snid 2487	. . . . . . . . 9 |- (/) e. {(/)}
35	32, 34	syl6eqel 1603	. . . . . . . 8 |- ((KFX) = (/) -> (KFX) e. {(/)})
36	35	necon3bi 1654	. . . . . . 7 |- (-. (KFX) e. {(/)} -> (KFX) =/= (/))
37	31, 36	syl 10	. . . . . 6 |- ((KFX) e. (P~A \ {(/)}) -> (KFX) =/= (/))
38	25, 37	syl 10	. . . . 5 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ (X e. A /\ K e. NN)) -> (KFX) =/= (/))
39	38	adantll 401	. . . 4 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> (KFX) =/= (/))
40	21, 22, 30, 39	syl3anc 870	. . 3 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> U.{v e. (KFX) | A.u e. (KFX) -. urv} e. (KFX))
41	20, 40	eqeltrd 1595	. 2 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> (XSK) e. (KFX))
42	30, 41	sseldd 2119	. 2 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> (XSK) e. A)
43	41, 42	jca 295	1 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> ((XSK) e. (KFX) /\ (XSK) e. A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 230   = wceq 997   e. wcel 999   =/= wne 1632  A.wral 1692  {crab 1695  Vcvv 1858   \ cdif 2095   u. cun 2096   (_ wss 2098  (/)c0 2331  P~cpw 2453  {csn 2461  <.cop 2463  U.cuni 2557   class class class wbr 2674  Icid 2887   We wwe 2973   X. cxp 3225   |` cres 3229  -->wf 3235  (class class class)co 4021  {copab2 4022  1c1 5300  NNcn 5361   seq1 cseq1 6566

This theorem is referenced by:  acdc2lem2 7581

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-reu 1698  df-rab 1699  df-v 1859  df-sbc 1989  df-csb 2052  df-dif 2100