Theorem acdc5lem1 7469

Description: Lemma for acdc5 7471.

Hypotheses

Ref	Expression
acdc5lem.1	|- A e. V
acdc5lem.2	|- S = {<.<.x, y>., z>. | ((x e. A /\ y e. NN) /\ z = U.{v e. (yFx) | A.u e. (yFx) -. urv})}
acdc5lem.3	|- G = (S seq1 ({<.1, c>.} u. (I |` (NN \ {1}))))

Assertion

Ref	Expression
acdc5lem1	|- (((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 acdc5lem1

Step	Hyp	Ref	Expression
1		oprex 3980	. . . . . . 7 |- (KFX) e. V
2	1	rabex 2722	. . . . . 6 |- {v e. (KFX) | A.u e. (KFX) -. urv} e. V
3	2	uniex 2867	. . . . 5 |- U.{v e. (KFX) | A.u e. (KFX) -. urv} e. V
4		opreq2 3966	. . . . . . 7 |- (x = X -> (yFx) = (yFX))
5		rabeq 1807	. . . . . . . 8 |- ((yFx) = (yFX) -> {v e. (yFx) | A.u e. (yFx) -. urv} = {v e. (yFX) | A.u e. (yFx) -. urv})
6		raleq1 1785	. . . . . . . . 9 |- ((yFx) = (yFX) -> (A.u e. (yFx) -. urv <-> A.u e. (yFX) -. urv))
7	6	rabbisdv 1805	. . . . . . . 8 |- ((yFx) = (yFX) -> {v e. (yFX) | A.u e. (yFx) -. urv} = {v e. (yFX) | A.u e. (yFX) -. urv})
8	5, 7	eqtrd 1506	. . . . . . 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 2509	. . . . 5 |- (x = X -> U.{v e. (yFx) | A.u e. (yFx) -. urv} = U.{v e. (yFX) | A.u e. (yFX) -. urv})
11		opreq1 3965	. . . . . . 7 |- (y = K -> (yFX) = (KFX))
12		rabeq 1807	. . . . . . . 8 |- ((yFX) = (KFX) -> {v e. (yFX) | A.u e. (yFX) -. urv} = {v e. (KFX) | A.u e. (yFX) -. urv})
13		raleq1 1785	. . . . . . . . 9 |- ((yFX) = (KFX) -> (A.u e. (yFX) -. urv <-> A.u e. (KFX) -. urv))
14	13	rabbisdv 1805	. . . . . . . 8 |- ((yFX) = (KFX) -> {v e. (KFX) | A.u e. (yFX) -. urv} = {v e. (KFX) | A.u e. (KFX) -. urv})
15	12, 14	eqtrd 1506	. . . . . . 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 2509	. . . . 5 |- (y = K -> U.{v e. (yFX) | A.u e. (yFX) -. urv} = U.{v e. (KFX) | A.u e. (KFX) -. urv})
18		acdc5lem.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 4025	. . . 4 |- ((X e. A /\ K e. NN) -> (XSK) = U.{v e. (KFX) | A.u e. (KFX) -. urv})
20	19	adantl 388	. . 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 2943	. . . . . 6 |- ((r We A /\ (KFX) (_ A /\ (KFX) =/= (/)) -> U.{v e. (KFX) | A.u e. (KFX) -. urv} e. (KFX))
22	21	3expb 833	. . . . 5 |- ((r We A /\ ((KFX) (_ A /\ (KFX) =/= (/))) -> U.{v e. (KFX) | A.u e. (KFX) -. urv} e. (KFX))
23		foprrn 4032	. . . . . . . . 9 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ K e. NN /\ X e. A) -> (KFX) e. (P~A \ {(/)}))
24		eldifsn 2460	. . . . . . . . 9 |- ((KFX) e. (P~A \ {(/)}) <-> ((KFX) e. P~A /\ (KFX) =/= (/)))
25	23, 24	sylib 198	. . . . . . . 8 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ K e. NN /\ X e. A) -> ((KFX) e. P~A /\ (KFX) =/= (/)))
26		elpwi 2404	. . . . . . . . 9 |- ((KFX) e. P~A -> (KFX) (_ A)
27	26	anim1i 334	. . . . . . . 8 |- (((KFX) e. P~A /\ (KFX) =/= (/)) -> ((KFX) (_ A /\ (KFX) =/= (/)))
28	25, 27	syl 10	. . . . . . 7 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ K e. NN /\ X e. A) -> ((KFX) (_ A /\ (KFX) =/= (/)))
29	28	3com23 838	. . . . . 6 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ X e. A /\ K e. NN) -> ((KFX) (_ A /\ (KFX) =/= (/)))
30	29	3expb 833	. . . . 5 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ (X e. A /\ K e. NN)) -> ((KFX) (_ A /\ (KFX) =/= (/)))
31	22, 30	sylan2 451	. . . 4 |- ((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))
32	31	anassrs 441	. . 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))
33	20, 32	eqeltrd 1547	. 2 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> (XSK) e. (KFX))
34	30	pm3.26d 321	. . . 4 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ (X e. A /\ K e. NN)) -> (KFX) (_ A)
35	34	adantll 392	. . 3 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> (KFX) (_ A)
36	35, 33	sseldd 2066	. 2 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> (XSK) e. A)
37	33, 36	jca 288	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 223   /\ w3a 774   = wceq 955   e. wcel 957   =/= wne 1584  A.wral 1644  {crab 1647  Vcvv 1809   \ cdif 2042   u. cun 2043   (_ wss 2045  (/)c0 2278  P~cpw 2399  {csn 2407  <.cop 2409  U.cuni 2500   class class class wbr 2616  Icid 2828   We wwe 2913   X. cxp 3165   |` cres 3169  -->wf 3175  (class class class)co 3960  {copab2 3961  1c1 5222  NNcn 5283   seq1 cseq1 6262

This theorem is referenced by:  acdc5lem2 7470

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv