Theorem acdc5lem1 7492

Description: Lemma for acdc5 7494.

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 3989	. . . . . . 7 |- (KFX) e. V
2	1	rabex 2730	. . . . . 6 |- {v e. (KFX) | A.u e. (KFX) -. urv} e. V
3	2	uniex 2876	. . . . 5 |- U.{v e. (KFX) | A.u e. (KFX) -. urv} e. V
4		opreq2 3975	. . . . . . 7 |- (x = X -> (yFx) = (yFX))
5		rabeq 1812	. . . . . . . 8 |- ((yFx) = (yFX) -> {v e. (yFx) | A.u e. (yFx) -. urv} = {v e. (yFX) | A.u e. (yFx) -. urv})
6		raleq1 1789	. . . . . . . . 9 |- ((yFx) = (yFX) -> (A.u e. (yFx) -. urv <-> A.u e. (yFX) -. urv))
7	6	rabbisdv 1810	. . . . . . . 8 |- ((yFx) = (yFX) -> {v e. (yFX) | A.u e. (yFx) -. urv} = {v e. (yFX) | A.u e. (yFX) -. urv})
8	5, 7	eqtrd 1510	. . . . . . 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 2516	. . . . 5 |- (x = X -> U.{v e. (yFx) | A.u e. (yFx) -. urv} = U.{v e. (yFX) | A.u e. (yFX) -. urv})
11		opreq1 3974	. . . . . . 7 |- (y = K -> (yFX) = (KFX))
12		rabeq 1812	. . . . . . . 8 |- ((yFX) = (KFX) -> {v e. (yFX) | A.u e. (yFX) -. urv} = {v e. (KFX) | A.u e. (yFX) -. urv})
13		raleq1 1789	. . . . . . . . 9 |- ((yFX) = (KFX) -> (A.u e. (yFX) -. urv <-> A.u e. (KFX) -. urv))
14	13	rabbisdv 1810	. . . . . . . 8 |- ((yFX) = (KFX) -> {v e. (KFX) | A.u e. (yFX) -. urv} = {v e. (KFX) | A.u e. (KFX) -. urv})
15	12, 14	eqtrd 1510	. . . . . . 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 2516	. . . . 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 4034	. . . 4 |- ((X e. A /\ K e. NN) -> (XSK) = U.{v e. (KFX) | A.u e. (KFX) -. urv})
20	19	adantl 390	. . 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 2952	. . . . . 6 |- ((r We A /\ (KFX) (_ A /\ (KFX) =/= (/)) -> U.{v e. (KFX) | A.u e. (KFX) -. urv} e. (KFX))
22	21	3expb 836	. . . . 5 |- ((r We A /\ ((KFX) (_ A /\ (KFX) =/= (/))) -> U.{v e. (KFX) | A.u e. (KFX) -. urv} e. (KFX))
23		foprrn 4041	. . . . . . . . 9 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ K e. NN /\ X e. A) -> (KFX) e. (P~A \ {(/)}))
24		eldifsn 2466	. . . . . . . . 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 2410	. . . . . . . . 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 841	. . . . . 6 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ X e. A /\ K e. NN) -> ((KFX) (_ A /\ (KFX) =/= (/)))
30	29	3expb 836	. . . . 5 |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ (X e. A /\ K e. NN)) -> ((KFX) (_ A /\ (KFX) =/= (/)))
31	22, 30	sylan2 453	. . . 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 443	. . 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 1551	. 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 394	. . 3 |- (((r We A /\ F:(NN X. A)-->(P~A \ {(/)})) /\ (X e. A /\ K e. NN)) -> (KFX) (_ A)
36	35, 33	sseldd 2071	. 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 777   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  {crab 1651  Vcvv 1814   \ cdif 2047   u. cun 2048   (_ wss 2050  (/)c0 2283  P~cpw 2405  {csn 2413  <.cop 2415  U.cuni 2507   class class class wbr 2624  Icid 2837   We wwe 2922   X. cxp 3174   |` cres 3178  -->wf 3184  (class class class)co 3969  {copab2 3970  1c1 5247  NNcn 5308   seq1 cseq1 6308

This theorem is referenced by:  acdc5lem2 7493

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200