Theorem fun11 4480

Description: Two ways of stating that A is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un₂ (A)" for one-to-one (but not necessarily a function).

Assertion

Ref	Expression
fun11	|- ((Fun `'`'A /\ Fun `'A) <-> A.xA.yA.zA.w((xAy /\ zAw) -> (x = z <-> y = w)))

Distinct variable group:   x,y,z,w,A

Proof of Theorem fun11

Step	Hyp	Ref	Expression
1		dfbi2 572	. . . . . . . 8 |- ((x = z <-> y = w) <-> ((x = z -> y = w) /\ (y = w -> x = z)))
2	1	imbi2i 202	. . . . . . 7 |- (((xAy /\ zAw) -> (x = z <-> y = w)) <-> ((xAy /\ zAw) -> ((x = z -> y = w) /\ (y = w -> x = z))))
3		pm4.76 660	. . . . . . 7 |- ((((xAy /\ zAw) -> (x = z -> y = w)) /\ ((xAy /\ zAw) -> (y = w -> x = z))) <-> ((xAy /\ zAw) -> ((x = z -> y = w) /\ (y = w -> x = z))))
4		bi2.04 177	. . . . . . . 8 |- (((xAy /\ zAw) -> (x = z -> y = w)) <-> (x = z -> ((xAy /\ zAw) -> y = w)))
5		bi2.04 177	. . . . . . . 8 |- (((xAy /\ zAw) -> (y = w -> x = z)) <-> (y = w -> ((xAy /\ zAw) -> x = z)))
6	4, 5	anbi12i 540	. . . . . . 7 |- ((((xAy /\ zAw) -> (x = z -> y = w)) /\ ((xAy /\ zAw) -> (y = w -> x = z))) <-> ((x = z -> ((xAy /\ zAw) -> y = w)) /\ (y = w -> ((xAy /\ zAw) -> x = z))))
7	2, 3, 6	3bitr2i 196	. . . . . 6 |- (((xAy /\ zAw) -> (x = z <-> y = w)) <-> ((x = z -> ((xAy /\ zAw) -> y = w)) /\ (y = w -> ((xAy /\ zAw) -> x = z))))
8	7	2albii 1347	. . . . 5 |- (A.xA.y((xAy /\ zAw) -> (x = z <-> y = w)) <-> A.xA.y((x = z -> ((xAy /\ zAw) -> y = w)) /\ (y = w -> ((xAy /\ zAw) -> x = z))))
9		19.26-2 1417	. . . . 5 |- (A.xA.y((x = z -> ((xAy /\ zAw) -> y = w)) /\ (y = w -> ((xAy /\ zAw) -> x = z))) <-> (A.xA.y(x = z -> ((xAy /\ zAw) -> y = w)) /\ A.xA.y(y = w -> ((xAy /\ zAw) -> x = z))))
10		alcom 1379	. . . . . . 7 |- (A.xA.y(x = z -> ((xAy /\ zAw) -> y = w)) <-> A.yA.x(x = z -> ((xAy /\ zAw) -> y = w)))
11		ax-17 1317	. . . . . . . . 9 |- (((zAy /\ zAw) -> y = w) -> A.x((zAy /\ zAw) -> y = w))
12		breq1 3341	. . . . . . . . . . 11 |- (x = z -> (xAy <-> zAy))
13	12	anbi1d 679	. . . . . . . . . 10 |- (x = z -> ((xAy /\ zAw) <-> (zAy /\ zAw)))
14	13	imbi1d 675	. . . . . . . . 9 |- (x = z -> (((xAy /\ zAw) -> y = w) <-> ((zAy /\ zAw) -> y = w)))
15	11, 14	equsal 1511	. . . . . . . 8 |- (A.x(x = z -> ((xAy /\ zAw) -> y = w)) <-> ((zAy /\ zAw) -> y = w))
16	15	albii 1346	. . . . . . 7 |- (A.yA.x(x = z -> ((xAy /\ zAw) -> y = w)) <-> A.y((zAy /\ zAw) -> y = w))
17	10, 16	bitri 190	. . . . . 6 |- (A.xA.y(x = z -> ((xAy /\ zAw) -> y = w)) <-> A.y((zAy /\ zAw) -> y = w))
18		ax-17 1317	. . . . . . . 8 |- (((xAw /\ zAw) -> x = z) -> A.y((xAw /\ zAw) -> x = z))
19		breq2 3342	. . . . . . . . . 10 |- (y = w -> (xAy <-> xAw))
20	19	anbi1d 679	. . . . . . . . 9 |- (y = w -> ((xAy /\ zAw) <-> (xAw /\ zAw)))
21	20	imbi1d 675	. . . . . . . 8 |- (y = w -> (((xAy /\ zAw) -> x = z) <-> ((xAw /\ zAw) -> x = z)))
22	18, 21	equsal 1511	. . . . . . 7 |- (A.y(y = w -> ((xAy /\ zAw) -> x = z)) <-> ((xAw /\ zAw) -> x = z))
23	22	albii 1346	. . . . . 6 |- (A.xA.y(y = w -> ((xAy /\ zAw) -> x = z)) <-> A.x((xAw /\ zAw) -> x = z))
24	17, 23	anbi12i 540	. . . . 5 |- ((A.xA.y(x = z -> ((xAy /\ zAw) -> y = w)) /\ A.xA.y(y = w -> ((xAy /\ zAw) -> x = z))) <-> (A.y((zAy /\ zAw) -> y = w) /\ A.x((xAw /\ zAw) -> x = z)))
25	8, 9, 24	3bitri 194	. . . 4 |- (A.xA.y((xAy /\ zAw) -> (x = z <-> y = w)) <-> (A.y((zAy /\ zAw) -> y = w) /\ A.x((xAw /\ zAw) -> x = z)))
26	25	2albii 1347	. . 3 |- (A.zA.wA.xA.y((xAy /\ zAw) -> (x = z <-> y = w)) <-> A.zA.w(A.y((zAy /\ zAw) -> y = w) /\ A.x((xAw /\ zAw) -> x = z)))
27		19.26-2 1417	. . 3 |- (A.zA.w(A.y((zAy /\ zAw) -> y = w) /\ A.x((xAw /\ zAw) -> x = z)) <-> (A.zA.wA.y((zAy /\ zAw) -> y = w) /\ A.zA.wA.x((xAw /\ zAw) -> x = z)))
28	26, 27	bitr2i 191	. 2 |- ((A.zA.wA.y((zAy /\ zAw) -> y = w) /\ A.zA.wA.x((xAw /\ zAw) -> x = z)) <-> A.zA.wA.xA.y((xAy /\ zAw) -> (x = z <-> y = w)))
29		fun2cnv 4477	. . . 4 |- (Fun `'`'A <-> A.zE*y zAy)
30		breq2 3342	. . . . . 6 |- (y = w -> (zAy <-> zAw))
31	30	mo4 1799	. . . . 5 |- (E*y zAy <-> A.yA.w((zAy /\ zAw) -> y = w))
32	31	albii 1346	. . . 4 |- (A.zE*y zAy <-> A.zA.yA.w((zAy /\ zAw) -> y = w))
33		alcom 1379	. . . . 5 |- (A.yA.w((zAy /\ zAw) -> y = w) <-> A.wA.y((zAy /\ zAw) -> y = w))
34	33	albii 1346	. . . 4 |- (A.zA.yA.w((zAy /\ zAw) -> y = w) <-> A.zA.wA.y((zAy /\ zAw) -> y = w))
35	29, 32, 34	3bitri 194	. . 3 |- (Fun `'`'A <-> A.zA.wA.y((zAy /\ zAw) -> y = w))
36		funcnv2 4474	. . . 4 |- (Fun `'A <-> A.wE*x xAw)
37		breq1 3341	. . . . . 6 |- (x = z -> (xAw <-> zAw))
38	37	mo4 1799	. . . . 5 |- (E*x xAw <-> A.xA.z((xAw /\ zAw) -> x = z))
39	38	albii 1346	. . . 4 |- (A.wE*x xAw <-> A.wA.xA.z((xAw /\ zAw) -> x = z))
40		alcom 1379	. . . . . 6 |- (A.xA.z((xAw /\ zAw) -> x = z) <-> A.zA.x((xAw /\ zAw) -> x = z))
41	40	albii 1346	. . . . 5 |- (A.wA.xA.z((xAw /\ zAw) -> x = z) <-> A.wA.zA.x((xAw /\ zAw) -> x = z))
42		alcom 1379	. . . . 5 |- (A.wA.zA.x((xAw /\ zAw) -> x = z) <-> A.zA.wA.x((xAw /\ zAw) -> x = z))
43	41, 42	bitri 190	. . . 4 |- (A.wA.xA.z((xAw /\ zAw) -> x = z) <-> A.zA.wA.x((xAw /\ zAw) -> x = z))
44	36, 39, 43	3bitri 194	. . 3 |- (Fun `'A <-> A.zA.wA.x((xAw /\ zAw) -> x = z))
45	35, 44	anbi12i 540	. 2 |- ((Fun `'`'A /\ Fun `'A) <-> (A.zA.wA.y((zAy /\ zAw) -> y = w) /\ A.zA.wA.x((xAw /\ zAw) -> x = z)))
46		alrot4 1451	. 2 |- (A.xA.yA.zA.w((xAy /\ zAw) -> (x = z <-> y = w)) <-> A.zA.wA.xA.y((xAy /\ zAw) -> (x = z <-> y = w)))
47	28, 45, 46	3bitr4i 200	1 |- ((Fun `'`'A /\ Fun `'A) <-> A.xA.yA.zA.w((xAy /\ zAw) -> (x = z <-> y = w)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E*wmo 1772   class class class wbr 3338  `'ccnv 3985  Fun wfun 3992

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-fun 4008

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