Theorem rcfpfillem6 14933

Description: Lemma for rcfpfil 14934.

Assertion

Ref	Expression
rcfpfillem6	|- ((u e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))} /\ v C_ A /\ u C_ v) -> v e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))})

Distinct variable group:   A,b,u,v,x

Proof of Theorem rcfpfillem6

Step	Hyp	Ref	Expression
1		visset 2295	. . . . 5 |- u e. _V
2		rcfpfillem1 14928	. . . . . 6 |- (u e. _V -> (u e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))} <-> E.b(b C_ A /\ b e. Fin /\ u = (A \ b))))
3		sseq1 2637	. . . . . . . 8 |- (b = c -> (b C_ A <-> c C_ A))
4		eleq1 1957	. . . . . . . 8 |- (b = c -> (b e. Fin <-> c e. Fin))
5		difeq2 2719	. . . . . . . . 9 |- (b = c -> (A \ b) = (A \ c))
6	5	eqeq2d 1895	. . . . . . . 8 |- (b = c -> (u = (A \ b) <-> u = (A \ c)))
7	3, 4, 6	3anbi123d 1168	. . . . . . 7 |- (b = c -> ((b C_ A /\ b e. Fin /\ u = (A \ b)) <-> (c C_ A /\ c e. Fin /\ u = (A \ c))))
8	7	cbvexv 1697	. . . . . 6 |- (E.b(b C_ A /\ b e. Fin /\ u = (A \ b)) <-> E.c(c C_ A /\ c e. Fin /\ u = (A \ c)))
9	2, 8	syl6bb 595	. . . . 5 |- (u e. _V -> (u e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))} <-> E.c(c C_ A /\ c e. Fin /\ u = (A \ c))))
10	1, 9	ax-mp 7	. . . 4 |- (u e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))} <-> E.c(c C_ A /\ c e. Fin /\ u = (A \ c)))
11		ssdifss 2736	. . . . . . . . 9 |- (c C_ A -> (c \ (v \ u)) C_ A)
12	11	3ad2ant1 897	. . . . . . . 8 |- ((c C_ A /\ c e. Fin /\ u = (A \ c)) -> (c \ (v \ u)) C_ A)
13	12	3ad2ant1 897	. . . . . . 7 |- (((c C_ A /\ c e. Fin /\ u = (A \ c)) /\ v C_ A /\ u C_ v) -> (c \ (v \ u)) C_ A)
14		difss 2735	. . . . . . . . . 10 |- (c \ (v \ u)) C_ c
15		ssfi 5630	. . . . . . . . . 10 |- ((c e. Fin /\ (c \ (v \ u)) C_ c) -> (c \ (v \ u)) e. Fin)
16	14, 15	mpan2 760	. . . . . . . . 9 |- (c e. Fin -> (c \ (v \ u)) e. Fin)
17	16	3ad2ant2 898	. . . . . . . 8 |- ((c C_ A /\ c e. Fin /\ u = (A \ c)) -> (c \ (v \ u)) e. Fin)
18	17	3ad2ant1 897	. . . . . . 7 |- (((c C_ A /\ c e. Fin /\ u = (A \ c)) /\ v C_ A /\ u C_ v) -> (c \ (v \ u)) e. Fin)
19		dfss4 2827	. . . . . . . . . . . . . . . . . . 19 |- (c C_ A <-> (A \ (A \ c)) = c)
20		difeq2 2719	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (u = (A \ c) -> (A \ u) = (A \ (A \ c)))
21	20	eqcomd 1889	. . . . . . . . . . . . . . . . . . . . . . 23 |- (u = (A \ c) -> (A \ (A \ c)) = (A \ u))
22	21	eqeq1d 1892	. . . . . . . . . . . . . . . . . . . . . 22 |- (u = (A \ c) -> ((A \ (A \ c)) = c <-> (A \ u) = c))
23	22	biimpd 170	. . . . . . . . . . . . . . . . . . . . 21 |- (u = (A \ c) -> ((A \ (A \ c)) = c -> (A \ u) = c))
24		undif 2954	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (u C_ v <-> (u u. (v \ u)) = v)
25		difeq2 2719	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((u u. (v \ u)) = v -> (A \ (u u. (v \ u))) = (A \ v))
26	24, 25	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (u C_ v -> (A \ (u u. (v \ u))) = (A \ v))
27		difun1 2853	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (A \ (u u. (v \ u))) = ((A \ u) \ (v \ u))
28	26, 27	syl5eqr 1942	. . . . . . . . . . . . . . . . . . . . . . 23 |- (u C_ v -> ((A \ u) \ (v \ u)) = (A \ v))
29	28	eqeq1d 1892	. . . . . . . . . . . . . . . . . . . . . 22 |- (u C_ v -> (((A \ u) \ (v \ u)) = (c \ (v \ u)) <-> (A \ v) = (c \ (v \ u))))
30		difeq1 2717	. . . . . . . . . . . . . . . . . . . . . 22 |- ((A \ u) = c -> ((A \ u) \ (v \ u)) = (c \ (v \ u)))
31	29, 30	syl5cbi 226	. . . . . . . . . . . . . . . . . . . . 21 |- ((A \ u) = c -> (u C_ v -> (A \ v) = (c \ (v \ u))))
32	23, 31	syl6com 64	. . . . . . . . . . . . . . . . . . . 20 |- ((A \ (A \ c)) = c -> (u = (A \ c) -> (u C_ v -> (A \ v) = (c \ (v \ u)))))
33	32	a1d 15	. . . . . . . . . . . . . . . . . . 19 |- ((A \ (A \ c)) = c -> (c e. Fin -> (u = (A \ c) -> (u C_ v -> (A \ v) = (c \ (v \ u))))))
34	19, 33	sylbi 216	. . . . . . . . . . . . . . . . . 18 |- (c C_ A -> (c e. Fin -> (u = (A \ c) -> (u C_ v -> (A \ v) = (c \ (v \ u))))))
35	34	3imp 1061	. . . . . . . . . . . . . . . . 17 |- ((c C_ A /\ c e. Fin /\ u = (A \ c)) -> (u C_ v -> (A \ v) = (c \ (v \ u))))
36	35	impcom 378	. . . . . . . . . . . . . . . 16 |- ((u C_ v /\ (c C_ A /\ c e. Fin /\ u = (A \ c))) -> (A \ v) = (c \ (v \ u)))
37	36	difeq2d 2726	. . . . . . . . . . . . . . 15 |- ((u C_ v /\ (c C_ A /\ c e. Fin /\ u = (A \ c))) -> (A \ (A \ v)) = (A \ (c \ (v \ u))))
38	37	eqeq2d 1895	. . . . . . . . . . . . . 14 |- ((u C_ v /\ (c C_ A /\ c e. Fin /\ u = (A \ c))) -> (v = (A \ (A \ v)) <-> v = (A \ (c \ (v \ u)))))
39	38	biimpd 170	. . . . . . . . . . . . 13 |- ((u C_ v /\ (c C_ A /\ c e. Fin /\ u = (A \ c))) -> (v = (A \ (A \ v)) -> v = (A \ (c \ (v \ u)))))
40	39	ex 402	. . . . . . . . . . . 12 |- (u C_ v -> ((c C_ A /\ c e. Fin /\ u = (A \ c)) -> (v = (A \ (A \ v)) -> v = (A \ (c \ (v \ u))))))
41	40	com13 37	. . . . . . . . . . 11 |- (v = (A \ (A \ v)) -> ((c C_ A /\ c e. Fin /\ u = (A \ c)) -> (u C_ v -> v = (A \ (c \ (v \ u))))))
42	41	eqcoms 1887	. . . . . . . . . 10 |- ((A \ (A \ v)) = v -> ((c C_ A /\ c e. Fin /\ u = (A \ c)) -> (u C_ v -> v = (A \ (c \ (v \ u))))))
43	42	com12 14	. . . . . . . . 9 |- ((c C_ A /\ c e. Fin /\ u = (A \ c)) -> ((A \ (A \ v)) = v -> (u C_ v -> v = (A \ (c \ (v \ u))))))
44		dfss4 2827	. . . . . . . . 9 |- (v C_ A <-> (A \ (A \ v)) = v)
45	43, 44	syl5ib 223	. . . . . . . 8 |- ((c C_ A /\ c e. Fin /\ u = (A \ c)) -> (v C_ A -> (u C_ v -> v = (A \ (c \ (v \ u))))))
46	45	3imp 1061	. . . . . . 7 |- (((c C_ A /\ c e. Fin /\ u = (A \ c)) /\ v C_ A /\ u C_ v) -> v = (A \ (c \ (v \ u))))
47		visset 2295	. . . . . . . . 9 |- c e. _V
48		difexg 3458	. . . . . . . . 9 |- (c e. _V -> (c \ (v \ u)) e. _V)
49	47, 48	ax-mp 7	. . . . . . . 8 |- (c \ (v \ u)) e. _V
50		sseq1 2637	. . . . . . . . 9 |- (b = (c \ (v \ u)) -> (b C_ A <-> (c \ (v \ u)) C_ A))
51		eleq1 1957	. . . . . . . . 9 |- (b = (c \ (v \ u)) -> (b e. Fin <-> (c \ (v \ u)) e. Fin))
52		difeq2 2719	. . . . . . . . . 10 |- (b = (c \ (v \ u)) -> (A \ b) = (A \ (c \ (v \ u))))
53	52	eqeq2d 1895	. . . . . . . . 9 |- (b = (c \ (v \ u)) -> (v = (A \ b) <-> v = (A \ (c \ (v \ u)))))
54	50, 51, 53	3anbi123d 1168	. . . . . . . 8 |- (b = (c \ (v \ u)) -> ((b C_ A /\ b e. Fin /\ v = (A \ b)) <-> ((c \ (v \ u)) C_ A /\ (c \ (v \ u)) e. Fin /\ v = (A \ (c \ (v \ u))))))
55	49, 54	cla4ev 2371	. . . . . . 7 |- (((c \ (v \ u)) C_ A /\ (c \ (v \ u)) e. Fin /\ v = (A \ (c \ (v \ u)))) -> E.b(b C_ A /\ b e. Fin /\ v = (A \ b)))
56	13, 18, 46, 55	syl111anc 1100	. . . . . 6 |- (((c C_ A /\ c e. Fin /\ u = (A \ c)) /\ v C_ A /\ u C_ v) -> E.b(b C_ A /\ b e. Fin /\ v = (A \ b)))
57	56	3exp 1066	. . . . 5 |- ((c C_ A /\ c e. Fin /\ u = (A \ c)) -> (v C_ A -> (u C_ v -> E.b(b C_ A /\ b e. Fin /\ v = (A \ b)))))
58	57	19.23aiv 1674	. . . 4 |- (E.c(c C_ A /\ c e. Fin /\ u = (A \ c)) -> (v C_ A -> (u C_ v -> E.b(b C_ A /\ b e. Fin /\ v = (A \ b)))))
59	10, 58	sylbi 216	. . 3 |- (u e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))} -> (v C_ A -> (u C_ v -> E.b(b C_ A /\ b e. Fin /\ v = (A \ b)))))
60	59	3imp 1061	. 2 |- ((u e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))} /\ v C_ A /\ u C_ v) -> E.b(b C_ A /\ b e. Fin /\ v = (A \ b)))
61		visset 2295	. . 3 |- v e. _V
62		eqeq1 1890	. . . . 5 |- (x = v -> (x = (A \ b) <-> v = (A \ b)))
63	62	3anbi3d 1174	. . . 4 |- (x = v -> ((b C_ A /\ b e. Fin /\ x = (A \ b)) <-> (b C_ A /\ b e. Fin /\ v = (A \ b))))
64	63	exbidv 1657	. . 3 |- (x = v -> (E.b(b C_ A /\ b e. Fin /\ x = (A \ b)) <-> E.b(b C_ A /\ b e. Fin /\ v = (A \ b))))
65	61, 64	elab 2403	. 2 |- (v e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))} <-> E.b(b C_ A /\ b e. Fin /\ v = (A \ b)))
66	60, 65	sylibr 217	1 |- ((u e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))} /\ v C_ A /\ u C_ v) -> v e. {x | E.b(b C_ A /\ b e. Fin /\ x = (A \ b))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  _Vcvv 2292   \ cdif 2590   u. cun 2591   C_ wss 2593  Fincfn 5426

This theorem is referenced by:  rcfpfil 14934

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-13 1311  ax-14 1312  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  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-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-er 5318  df-en 5427  df-fin 5430

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