Theorem rcfpfillem3 10589

Description: Lemma for rcfpfil 10597.

Assertion

Ref	Expression
rcfpfillem3	|- (A e. B -> U.{x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))} = A)

Distinct variable groups:   A,b,x   B,b

Proof of Theorem rcfpfillem3

Step	Hyp	Ref	Expression
1		19.42v 1308	. . . . . . . 8 |- (E.b(y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) <-> (y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b))))
2	1	bicomi 172	. . . . . . 7 |- ((y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b))) <-> E.b(y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))))
3	2	exbii 1051	. . . . . 6 |- (E.x(y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b))) <-> E.xE.b(y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))))
4		excom 1046	. . . . . . 7 |- (E.xE.b(y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) <-> E.bE.x(y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))))
5		3simp3 790	. . . . . . . . . . . . 13 |- ((b (_ A /\ b e. Fin /\ x = (A \ b)) -> x = (A \ b))
6	5	adantl 388	. . . . . . . . . . . 12 |- ((y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) -> x = (A \ b))
7		pm3.26 319	. . . . . . . . . . . 12 |- ((y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) -> y e. x)
8	6, 7	jca 288	. . . . . . . . . . 11 |- ((y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) -> (x = (A \ b) /\ y e. x))
9		3simpa 785	. . . . . . . . . . . 12 |- ((b (_ A /\ b e. Fin /\ x = (A \ b)) -> (b (_ A /\ b e. Fin))
10	9	adantl 388	. . . . . . . . . . 11 |- ((y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) -> (b (_ A /\ b e. Fin))
11	8, 10	jca 288	. . . . . . . . . 10 |- ((y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) -> ((x = (A \ b) /\ y e. x) /\ (b (_ A /\ b e. Fin)))
12	11	19.22i 1040	. . . . . . . . 9 |- (E.x(y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) -> E.x((x = (A \ b) /\ y e. x) /\ (b (_ A /\ b e. Fin)))
13		19.40 1094	. . . . . . . . 9 |- (E.x((x = (A \ b) /\ y e. x) /\ (b (_ A /\ b e. Fin)) -> (E.x(x = (A \ b) /\ y e. x) /\ E.x(b (_ A /\ b e. Fin)))
14		difexg 2722	. . . . . . . . . . . 12 |- (A e. B -> (A \ b) e. V)
15		eleq2 1535	. . . . . . . . . . . . . 14 |- (x = (A \ b) -> (y e. x <-> y e. (A \ b)))
16	15	ceqsexgv 1888	. . . . . . . . . . . . 13 |- ((A \ b) e. V -> (E.x(x = (A \ b) /\ y e. x) <-> y e. (A \ b)))
17		eldifi 2162	. . . . . . . . . . . . 13 |- (y e. (A \ b) -> y e. A)
18	16, 17	syl6bi 214	. . . . . . . . . . . 12 |- ((A \ b) e. V -> (E.x(x = (A \ b) /\ y e. x) -> y e. A))
19	14, 18	syl 10	. . . . . . . . . . 11 |- (A e. B -> (E.x(x = (A \ b) /\ y e. x) -> y e. A))
20	19	com12 11	. . . . . . . . . 10 |- (E.x(x = (A \ b) /\ y e. x) -> (A e. B -> y e. A))
21	20	adantr 389	. . . . . . . . 9 |- ((E.x(x = (A \ b) /\ y e. x) /\ E.x(b (_ A /\ b e. Fin)) -> (A e. B -> y e. A))
22	12, 13, 21	3syl 20	. . . . . . . 8 |- (E.x(y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) -> (A e. B -> y e. A))
23	22	19.23aiv 1295	. . . . . . 7 |- (E.bE.x(y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) -> (A e. B -> y e. A))
24	4, 23	sylbi 199	. . . . . 6 |- (E.xE.b(y e. x /\ (b (_ A /\ b e. Fin /\ x = (A \ b))) -> (A e. B -> y e. A))
25	3, 24	sylbi 199	. . . . 5 |- (E.x(y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b))) -> (A e. B -> y e. A))
26	25	com12 11	. . . 4 |- (A e. B -> (E.x(y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b))) -> y e. A))
27		eleq2 1535	. . . . . . 7 |- (x = A -> (y e. x <-> y e. A))
28		eqeq1 1481	. . . . . . . . 9 |- (x = A -> (x = (A \ b) <-> A = (A \ b)))
29	28	3anbi3d 899	. . . . . . . 8 |- (x = A -> ((b (_ A /\ b e. Fin /\ x = (A \ b)) <-> (b (_ A /\ b e. Fin /\ A = (A \ b))))
30	29	exbidv 1279	. . . . . . 7 |- (x = A -> (E.b(b (_ A /\ b e. Fin /\ x = (A \ b)) <-> E.b(b (_ A /\ b e. Fin /\ A = (A \ b))))
31	27, 30	anbi12d 628	. . . . . 6 |- (x = A -> ((y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b))) <-> (y e. A /\ E.b(b (_ A /\ b e. Fin /\ A = (A \ b)))))
32		ancom 435	. . . . . . 7 |- ((y e. A /\ E.b(b (_ A /\ b e. Fin /\ A = (A \ b))) <-> (E.b(b (_ A /\ b e. Fin /\ A = (A \ b)) /\ y e. A))
33		0ex 2711	. . . . . . . 8 |- (/) e. V
34		0ss 2301	. . . . . . . . 9 |- (/) (_ A
35		emfin 10477	. . . . . . . . 9 |- (/) e. Fin
36		dif0 2335	. . . . . . . . . 10 |- (A \ (/)) = A
37	36	eqcomi 1479	. . . . . . . . 9 |- A = (A \ (/))
38	34, 35, 37	3pm3.2i 818	. . . . . . . 8 |- ((/) (_ A /\ (/) e. Fin /\ A = (A \ (/)))
39		sseq1 2082	. . . . . . . . . 10 |- (b = (/) -> (b (_ A <-> (/) (_ A))
40		eleq1 1534	. . . . . . . . . 10 |- (b = (/) -> (b e. Fin <-> (/) e. Fin))
41		difeq2 2154	. . . . . . . . . . 11 |- (b = (/) -> (A \ b) = (A \ (/)))
42	41	eqeq2d 1486	. . . . . . . . . 10 |- (b = (/) -> (A = (A \ b) <-> A = (A \ (/))))
43	39, 40, 42	3anbi123d 893	. . . . . . . . 9 |- (b = (/) -> ((b (_ A /\ b e. Fin /\ A = (A \ b)) <-> ((/) (_ A /\ (/) e. Fin /\ A = (A \ (/)))))
44	43	cla4egv 1863	. . . . . . . 8 |- ((/) e. V -> (((/) (_ A /\ (/) e. Fin /\ A = (A \ (/))) -> E.b(b (_ A /\ b e. Fin /\ A = (A \ b))))
45	33, 38, 44	mp2 43	. . . . . . 7 |- E.b(b (_ A /\ b e. Fin /\ A = (A \ b))
46	32, 45	mpbiran 728	. . . . . 6 |- ((y e. A /\ E.b(b (_ A /\ b e. Fin /\ A = (A \ b))) <-> y e. A)
47	31, 46	syl6bb 536	. . . . 5 |- (x = A -> ((y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b))) <-> y e. A))
48	47	cla4egv 1863	. . . 4 |- (A e. B -> (y e. A -> E.x(y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b)))))
49	26, 48	impbid 516	. . 3 |- (A e. B -> (E.x(y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b))) <-> y e. A))
50		eluniab 2513	. . 3 |- (y e. U.{x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))} <-> E.x(y e. x /\ E.b(b (_ A /\ b e. Fin /\ x = (A \ b))))
51	49, 50	syl5bb 532	. 2 |- (A e. B -> (y e. U.{x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))} <-> y e. A))
52	51	eqrdv 1473	1 |- (A e. B -> U.{x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))} = A)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  Vcvv 1811   \ cdif 2044   (_ wss 2047  (/)c0 2280  U.cuni 2503  Fincfn 4367

This theorem is referenced by:  rcfpfil 10597

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-en 4368  df-fin 4371

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