Theorem fun11 3578

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 517	. . . . . . . 8 |- ((x = z <-> y = w) <-> ((x = z -> y = w) /\ (y = w -> x = z)))
2	1	imbi2i 185	. . . . . . 7 |- (((xAy /\ zAw) -> (x = z <-> y = w)) <-> ((xAy /\ zAw) -> ((x = z -> y = w) /\ (y = w -> x = z))))
3		pm4.76 602	. . . . . . 7 |- ((((xAy /\ zAw) -> (x = z -> y = w)) /\ ((xAy /\ zAw) -> (y = w -> x = z))) <-> ((xAy /\ zAw) -> ((x = z -> y = w) /\ (y = w -> x = z))))
4		bi2.04 160	. . . . . . . 8 |- (((xAy /\ zAw) -> (x = z -> y = w)) <-> (x = z -> ((xAy /\ zAw) -> y = w)))
5		bi2.04 160	. . . . . . . 8 |- (((xAy /\ zAw) -> (y = w -> x = z)) <-> (y = w -> ((xAy /\ zAw) -> x = z)))
6	4, 5	anbi12i 485	. . . . . . 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 179	. . . . . 6 |- (((xAy /\ zAw) -> (x = z <-> y = w)) <-> ((x = z -> ((xAy /\ zAw) -> y = w)) /\ (y = w -> ((xAy /\ zAw) -> x = z))))
8	7	2albii 1004	. . . . 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 1072	. . . . 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 1036	. . . . . . 7 |- (A.xA.y(x = z -> ((xAy /\ zAw) -> y = w)) <-> A.yA.x(x = z -> ((xAy /\ zAw) -> y = w)))
11		ax-17 975	. . . . . . . . 9 |- (((zAy /\ zAw) -> y = w) -> A.x((zAy /\ zAw) -> y = w))
12		breq1 2637	. . . . . . . . . . 11 |- (x = z -> (xAy <-> zAy))
13	12	anbi1d 620	. . . . . . . . . 10 |- (x = z -> ((xAy /\ zAw) <-> (zAy /\ zAw)))
14	13	imbi1d 616	. . . . . . . . 9 |- (x = z -> (((xAy /\ zAw) -> y = w) <-> ((zAy /\ zAw) -> y = w)))
15	11, 14	equsal 1155	. . . . . . . 8 |- (A.x(x = z -> ((xAy /\ zAw) -> y = w)) <-> ((zAy /\ zAw) -> y = w))
16	15	albii 1003	. . . . . . 7 |- (A.yA.x(x = z -> ((xAy /\ zAw) -> y = w)) <-> A.y((zAy /\ zAw) -> y = w))
17	10, 16	bitri 173	. . . . . 6 |- (A.xA.y(x = z -> ((xAy /\ zAw) -> y = w)) <-> A.y((zAy /\ zAw) -> y = w))
18		ax-17 975	. . . . . . . 8 |- (((xAw /\ zAw) -> x = z) -> A.y((xAw /\ zAw) -> x = z))
19		breq2 2638	. . . . . . . . . 10 |- (y = w -> (xAy <-> xAw))
20	19	anbi1d 620	. . . . . . . . 9 |- (y = w -> ((xAy /\ zAw) <-> (xAw /\ zAw)))
21	20	imbi1d 616	. . . . . . . 8 |- (y = w -> (((xAy /\ zAw) -> x = z) <-> ((xAw /\ zAw) -> x = z)))
22	18, 21	equsal 1155	. . . . . . 7 |- (A.y(y = w -> ((xAy /\ zAw) -> x = z)) <-> ((xAw /\ zAw) -> x = z))
23	22	albii 1003	. . . . . 6 |- (A.xA.y(y = w -> ((xAy /\ zAw) -> x = z)) <-> A.x((xAw /\ zAw) -> x = z))
24	17, 23	anbi12i 485	. . . . 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 177	. . . 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 1004	. . 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 1072	. . 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 174	. 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 3575	. . . 4 |- (Fun `'`'A <-> A.zE*y zAy)
30		breq2 2638	. . . . . 6 |- (y = w -> (zAy <-> zAw))
31	30	mo4 1407	. . . . 5 |- (E*y zAy <-> A.yA.w((zAy /\ zAw) -> y = w))
32	31	albii 1003	. . . 4 |- (A.zE*y zAy <-> A.zA.yA.w((zAy /\ zAw) -> y = w))
33		alcom 1036	. . . . 5 |- (A.yA.w((zAy /\ zAw) -> y = w) <-> A.wA.y((zAy /\ zAw) -> y = w))
34	33	albii 1003	. . . 4 |- (A.zA.yA.w((zAy /\ zAw) -> y = w) <-> A.zA.wA.y((zAy /\ zAw) -> y = w))
35	29, 32, 34	3bitri 177	. . 3 |- (Fun `'`'A <-> A.zA.wA.y((zAy /\ zAw) -> y = w))
36		funcnv2 3572	. . . 4 |- (Fun `'A <-> A.wE*x xAw)
37		breq1 2637	. . . . . 6 |- (x = z -> (xAw <-> zAw))
38	37	mo4 1407	. . . . 5 |- (E*x xAw <-> A.xA.z((xAw /\ zAw) -> x = z))
39	38	albii 1003	. . . 4 |- (A.wE*x xAw <-> A.wA.xA.z((xAw /\ zAw) -> x = z))
40		alcom 1036	. . . . . 6 |- (A.xA.z((xAw /\ zAw) -> x = z) <-> A.zA.x((xAw /\ zAw) -> x = z))
41	40	albii 1003	. . . . 5 |- (A.wA.xA.z((xAw /\ zAw) -> x = z) <-> A.wA.zA.x((xAw /\ zAw) -> x = z))
42		alcom 1036	. . . . 5 |- (A.wA.zA.x((xAw /\ zAw) -> x = z) <-> A.zA.wA.x((xAw /\ zAw) -> x = z))
43	41, 42	bitri 173	. . . 4 |- (A.wA.xA.z((xAw /\ zAw) -> x = z) <-> A.zA.wA.x((xAw /\ zAw) -> x = z))
44	36, 39, 43	3bitri 177	. . 3 |- (Fun `'A <-> A.zA.wA.x((xAw /\ zAw)