Theorem inficlALT 15372

Description: Alternate form of inficl 15757 for use with Jeff Hankins's mathbox.

Assertion

Ref	Expression
inficlALT	|- (A e. B -> (A.x e. A A.y e. A (x i^i y) e. A <-> ( fi ` A) = A))

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

Proof of Theorem inficlALT

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . . . . . . . 10 |- (A e. B -> A.x A e. B)
2		hba1 1350	. . . . . . . . . 10 |- (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) -> A.xA.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A))
3	1, 2	hban 1356	. . . . . . . . 9 |- ((A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)) -> A.x(A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)))
4		ax-17 1317	. . . . . . . . 9 |- (y e. A -> A.x y e. A)
5		19.8a 1376	. . . . . . . . . . . 12 |- (y = |^|x -> E.y y = |^|x)
6		intex 3465	. . . . . . . . . . . . 13 |- (x =/= (/) <-> |^|x e. _V)
7		isset 2296	. . . . . . . . . . . . 13 |- (|^|x e. _V <-> E.y y = |^|x)
8	6, 7	bitr2i 191	. . . . . . . . . . . 12 |- (E.y y = |^|x <-> x =/= (/))
9	5, 8	sylib 215	. . . . . . . . . . 11 |- (y = |^|x -> x =/= (/))
10	9	3ad2ant3 899	. . . . . . . . . 10 |- ((x C_ A /\ x e. Fin /\ y = |^|x) -> x =/= (/))
11		simprl3 923	. . . . . . . . . . . 12 |- (((A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)) /\ ((x C_ A /\ x e. Fin /\ y = |^|x) /\ x =/= (/))) -> y = |^|x)
12		ax-4 1319	. . . . . . . . . . . . . . . . . 18 |- (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) -> ((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A))
13	12	com12 14	. . . . . . . . . . . . . . . . 17 |- ((x C_ A /\ x =/= (/) /\ x e. Fin) -> (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) -> |^|x e. A))
14	13	3com23 1074	. . . . . . . . . . . . . . . 16 |- ((x C_ A /\ x e. Fin /\ x =/= (/)) -> (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) -> |^|x e. A))
15	14	3expa 1067	. . . . . . . . . . . . . . 15 |- (((x C_ A /\ x e. Fin) /\ x =/= (/)) -> (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) -> |^|x e. A))
16	15	3adantl3 1034	. . . . . . . . . . . . . 14 |- (((x C_ A /\ x e. Fin /\ y = |^|x) /\ x =/= (/)) -> (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) -> |^|x e. A))
17	16	impcom 378	. . . . . . . . . . . . 13 |- ((A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) /\ ((x C_ A /\ x e. Fin /\ y = |^|x) /\ x =/= (/))) -> |^|x e. A)
18	17	adantll 428	. . . . . . . . . . . 12 |- (((A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)) /\ ((x C_ A /\ x e. Fin /\ y = |^|x) /\ x =/= (/))) -> |^|x e. A)
19	11, 18	eqeltrd 1971	. . . . . . . . . . 11 |- (((A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)) /\ ((x C_ A /\ x e. Fin /\ y = |^|x) /\ x =/= (/))) -> y e. A)
20	19	exp32 408	. . . . . . . . . 10 |- ((A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)) -> ((x C_ A /\ x e. Fin /\ y = |^|x) -> (x =/= (/) -> y e. A)))
21	10, 20	mpdi 59	. . . . . . . . 9 |- ((A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)) -> ((x C_ A /\ x e. Fin /\ y = |^|x) -> y e. A))
22	3, 4, 21	19.23ad 1415	. . . . . . . 8 |- ((A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)) -> (E.x(x C_ A /\ x e. Fin /\ y = |^|x) -> y e. A))
23		snssi 3129	. . . . . . . . . . . . 13 |- (y e. A -> {y} C_ A)
24		snfi 5491	. . . . . . . . . . . . . 14 |- {y} e. Fin
25	24	a1i 8	. . . . . . . . . . . . 13 |- (y e. A -> {y} e. Fin)
26		visset 2295	. . . . . . . . . . . . . . . 16 |- y e. _V
27	26	intsn 3252	. . . . . . . . . . . . . . 15 |- |^|{y} = y
28	27	eqcomi 1888	. . . . . . . . . . . . . 14 |- y = |^|{y}
29	28	a1i 8	. . . . . . . . . . . . 13 |- (y e. A -> y = |^|{y})
30	23, 25, 29	3jca 1050	. . . . . . . . . . . 12 |- (y e. A -> ({y} C_ A /\ {y} e. Fin /\ y = |^|{y}))
31	30	adantl 424	. . . . . . . . . . 11 |- ((A e. B /\ y e. A) -> ({y} C_ A /\ {y} e. Fin /\ y = |^|{y}))
32		snex 3492	. . . . . . . . . . . 12 |- {y} e. _V
33		sseq1 2637	. . . . . . . . . . . . 13 |- (x = {y} -> (x C_ A <-> {y} C_ A))
34		eleq1 1957	. . . . . . . . . . . . 13 |- (x = {y} -> (x e. Fin <-> {y} e. Fin))
35		inteq 3217	. . . . . . . . . . . . . 14 |- (x = {y} -> |^|x = |^|{y})
36	35	eqeq2d 1895	. . . . . . . . . . . . 13 |- (x = {y} -> (y = |^|x <-> y = |^|{y}))
37	33, 34, 36	3anbi123d 1168	. . . . . . . . . . . 12 |- (x = {y} -> ((x C_ A /\ x e. Fin /\ y = |^|x) <-> ({y} C_ A /\ {y} e. Fin /\ y = |^|{y})))
38	32, 37	cla4ev 2371	. . . . . . . . . . 11 |- (({y} C_ A /\ {y} e. Fin /\ y = |^|{y}) -> E.x(x C_ A /\ x e. Fin /\ y = |^|x))
39	31, 38	syl 12	. . . . . . . . . 10 |- ((A e. B /\ y e. A) -> E.x(x C_ A /\ x e. Fin /\ y = |^|x))
40	39	ex 402	. . . . . . . . 9 |- (A e. B -> (y e. A -> E.x(x C_ A /\ x e. Fin /\ y = |^|x)))
41	40	adantr 425	. . . . . . . 8 |- ((A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)) -> (y e. A -> E.x(x C_ A /\ x e. Fin /\ y = |^|x)))
42	22, 41	impbid 574	. . . . . . 7 |- ((A e. B /\ A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)) -> (E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A))
43	42	ex 402	. . . . . 6 |- (A e. B -> (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) -> (E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A)))
44	43	19.21adv 1666	. . . . 5 |- (A e. B -> (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) -> A.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A)))
45		hbe1 1363	. . . . . . . 8 |- (E.x(x C_ A /\ x e. Fin /\ y = |^|x) -> A.xE.x(x C_ A /\ x e. Fin /\ y = |^|x))
46	45, 4	hbbi 1357	. . . . . . 7 |- ((E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A) -> A.x(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A))
47	46	hbal 1352	. . . . . 6 |- (A.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A) -> A.xA.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A))
48		eqeq1 1890	. . . . . . . . . . . . . . . . 17 |- (y = |^|x -> (y = |^|z <-> |^|x = |^|z))
49	48	3anbi3d 1174	. . . . . . . . . . . . . . . 16 |- (y = |^|x -> ((z C_ A /\ z e. Fin /\ y = |^|z) <-> (z C_ A /\ z e. Fin /\ |^|x = |^|z)))
50	49	exbidv 1657	. . . . . . . . . . . . . . 15 |- (y = |^|x -> (E.z(z C_ A /\ z e. Fin /\ y = |^|z) <-> E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z)))
51		eleq1 1957	. . . . . . . . . . . . . . 15 |- (y = |^|x -> (y e. A <-> |^|x e. A))
52	50, 51	bibi12d 691	. . . . . . . . . . . . . 14 |- (y = |^|x -> ((E.z(z C_ A /\ z e. Fin /\ y = |^|z) <-> y e. A) <-> (E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) <-> |^|x e. A)))
53	52	cla4gv 2364	. . . . . . . . . . . . 13 |- (|^|x e. _V -> (A.y(E.z(z C_ A /\ z e. Fin /\ y = |^|z) <-> y e. A) -> (E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) <-> |^|x e. A)))
54	6, 53	sylbi 216	. . . . . . . . . . . 12 |- (x =/= (/) -> (A.y(E.z(z C_ A /\ z e. Fin /\ y = |^|z) <-> y e. A) -> (E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) <-> |^|x e. A)))
55	54	3ad2ant2 898	. . . . . . . . . . 11 |- ((x C_ A /\ x =/= (/) /\ x e. Fin) -> (A.y(E.z(z C_ A /\ z e. Fin /\ y = |^|z) <-> y e. A) -> (E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) <-> |^|x e. A)))
56	55	adantl 424	. . . . . . . . . 10 |- ((A e. B /\ (x C_ A /\ x =/= (/) /\ x e. Fin)) -> (A.y(E.z(z C_ A /\ z e. Fin /\ y = |^|z) <-> y e. A) -> (E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) <-> |^|x e. A)))
57		simp1 876	. . . . . . . . . . . . . 14 |- ((x C_ A /\ x =/= (/) /\ x e. Fin) -> x C_ A)
58	57	adantl 424	. . . . . . . . . . . . 13 |- ((A e. B /\ (x C_ A /\ x =/= (/) /\ x e. Fin)) -> x C_ A)
59		simp3 878	. . . . . . . . . . . . . 14 |- ((x C_ A /\ x =/= (/) /\ x e. Fin) -> x e. Fin)
60	59	adantl 424	. . . . . . . . . . . . 13 |- ((A e. B /\ (x C_ A /\ x =/= (/) /\ x e. Fin)) -> x e. Fin)
61		eqidd 1885	. . . . . . . . . . . . 13 |- ((A e. B /\ (x C_ A /\ x =/= (/) /\ x e. Fin)) -> |^|x = |^|x)
62	58, 60, 61	3jca 1050	. . . . . . . . . . . 12 |- ((A e. B /\ (x C_ A /\ x =/= (/) /\ x e. Fin)) -> (x C_ A /\ x e. Fin /\ |^|x = |^|x))
63		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
64		sseq1 2637	. . . . . . . . . . . . . 14 |- (z = x -> (z C_ A <-> x C_ A))
65		eleq1 1957	. . . . . . . . . . . . . 14 |- (z = x -> (z e. Fin <-> x e. Fin))
66		inteq 3217	. . . . . . . . . . . . . . 15 |- (z = x -> |^|z = |^|x)
67	66	eqeq2d 1895	. . . . . . . . . . . . . 14 |- (z = x -> (|^|x = |^|z <-> |^|x = |^|x))
68	64, 65, 67	3anbi123d 1168	. . . . . . . . . . . . 13 |- (z = x -> ((z C_ A /\ z e. Fin /\ |^|x = |^|z) <-> (x C_ A /\ x e. Fin /\ |^|x = |^|x)))
69	63, 68	cla4ev 2371	. . . . . . . . . . . 12 |- ((x C_ A /\ x e. Fin /\ |^|x = |^|x) -> E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z))
70		pm2.27 76	. . . . . . . . . . . 12 |- (E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) -> ((E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) -> |^|x e. A) -> |^|x e. A))
71	62, 69, 70	3syl 24	. . . . . . . . . . 11 |- ((A e. B /\ (x C_ A /\ x =/= (/) /\ x e. Fin)) -> ((E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) -> |^|x e. A) -> |^|x e. A))
72		bi1 165	. . . . . . . . . . 11 |- ((E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) <-> |^|x e. A) -> (E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) -> |^|x e. A))
73	71, 72	syl5 20	. . . . . . . . . 10 |- ((A e. B /\ (x C_ A /\ x =/= (/) /\ x e. Fin)) -> ((E.z(z C_ A /\ z e. Fin /\ |^|x = |^|z) <-> |^|x e. A) -> |^|x e. A))
74	56, 73	syld 30	. . . . . . . . 9 |- ((A e. B /\ (x C_ A /\ x =/= (/) /\ x e. Fin)) -> (A.y(E.z(z C_ A /\ z e. Fin /\ y = |^|z) <-> y e. A) -> |^|x e. A))
75		sseq1 2637	. . . . . . . . . . . . 13 |- (x = z -> (x C_ A <-> z C_ A))
76		eleq1 1957	. . . . . . . . . . . . 13 |- (x = z -> (x e. Fin <-> z e. Fin))
77		inteq 3217	. . . . . . . . . . . . . 14 |- (x = z -> |^|x = |^|z)
78	77	eqeq2d 1895	. . . . . . . . . . . . 13 |- (x = z -> (y = |^|x <-> y = |^|z))
79	75, 76, 78	3anbi123d 1168	. . . . . . . . . . . 12 |- (x = z -> ((x C_ A /\ x e. Fin /\ y = |^|x) <-> (z C_ A /\ z e. Fin /\ y = |^|z)))
80	79	cbvexv 1697	. . . . . . . . . . 11 |- (E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> E.z(z C_ A /\ z e. Fin /\ y = |^|z))
81	80	bibi1i 671	. . . . . . . . . 10 |- ((E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A) <-> (E.z(z C_ A /\ z e. Fin /\ y = |^|z) <-> y e. A))
82	81	albii 1346	. . . . . . . . 9 |- (A.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A) <-> A.y(E.z(z C_ A /\ z e. Fin /\ y = |^|z) <-> y e. A))
83	74, 82	syl5ib 223	. . . . . . . 8 |- ((A e. B /\ (x C_ A /\ x =/= (/) /\ x e. Fin)) -> (A.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A) -> |^|x e. A))
84	83	ex 402	. . . . . . 7 |- (A e. B -> ((x C_ A /\ x =/= (/) /\ x e. Fin) -> (A.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A) -> |^|x e. A)))
85	84	com23 36	. . . . . 6 |- (A e. B -> (A.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A) -> ((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)))
86	1, 47, 85	19.21ad 1406	. . . . 5 |- (A e. B -> (A.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A) -> A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A)))
87	44, 86	impbid 574	. . . 4 |- (A e. B -> (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) <-> A.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A)))
88		abeq1 2000	. . . 4 |- ({y | E.x(x C_ A /\ x e. Fin /\ y = |^|x)} = A <-> A.y(E.x(x C_ A /\ x e. Fin /\ y = |^|x) <-> y e. A))
89	87, 88	syl6bbr 597	. . 3 |- (A e. B -> (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) <-> {y | E.x(x C_ A /\ x e. Fin /\ y = |^|x)} = A))
90		fiv 10212	. . . 4 |- (A e. B -> ( fi ` A) = {y | E.x(x C_ A /\ x e. Fin /\ y = |^|x)})
91	90	eqeq1d 1892	. . 3 |- (A e. B -> (( fi ` A) = A <-> {y | E.x(x C_ A /\ x e. Fin /\ y = |^|x)} = A))
92	89, 91	bitr4d 590	. 2 |- (A e. B -> (A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A) <-> ( fi ` A) = A))
93		fiint 5650	. 2 |- (A.x e. A A.y e. A (x i^i y) e. A <-> A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A))
94	92, 93	syl5bb 591	1 |- (A e. B -> (A.x e. A A.y e. A (x i^i y) e. A <-> ( fi ` A) = A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  |^|cint 3214  ` cfv 3998  Fincfn 5426   fi cfi 10210

This theorem is referenced by:  fitop 15401  fmfnfmlem3 15596

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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-int 3215  df-iun 3257  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-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-er 5318  df-en 5427  df-fin 5430  df-fi 10211

Copyright terms: Public domain