Theorem fun11 5160

Description: Two ways of stating that A is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un₂ (A)" for one-to-one. (Contributed by NM, 17-Jan-2006.) (Revised by Scott Fenton, 18-Apr-2021.)

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 609	. . . . . . . 8 |- ((x = z <-> y = w) <-> ((x = z -> y = w) /\ (y = w -> x = z)))
2	1	imbi2i 303	. . . . . . 7 |- (((xAy /\ zAw) -> (x = z <-> y = w)) <-> ((xAy /\ zAw) -> ((x = z -> y = w) /\ (y = w -> x = z))))
3		pm4.76 836	. . . . . . 7 |- ((((xAy /\ zAw) -> (x = z -> y = w)) /\ ((xAy /\ zAw) -> (y = w -> x = z))) <-> ((xAy /\ zAw) -> ((x = z -> y = w) /\ (y = w -> x = z))))
4		bi2.04 350	. . . . . . . 8 |- (((xAy /\ zAw) -> (x = z -> y = w)) <-> (x = z -> ((xAy /\ zAw) -> y = w)))
5		bi2.04 350	. . . . . . . 8 |- (((xAy /\ zAw) -> (y = w -> x = z)) <-> (y = w -> ((xAy /\ zAw) -> x = z)))
6	4, 5	anbi12i 678	. . . . . . 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 264	. . . . . 6 |- (((xAy /\ zAw) -> (x = z <-> y = w)) <-> ((x = z -> ((xAy /\ zAw) -> y = w)) /\ (y = w -> ((xAy /\ zAw) -> x = z))))
8	7	2albii 1567	. . . . 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 1594	. . . . 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 1737	. . . . . . 7 |- (A.xA.y(x = z -> ((xAy /\ zAw) -> y = w)) <-> A.yA.x(x = z -> ((xAy /\ zAw) -> y = w)))
11		nfv 1619	. . . . . . . . 9 |- F/x((zAy /\ zAw) -> y = w)
12		breq1 4643	. . . . . . . . . . 11 |- (x = z -> (xAy <-> zAy))
13	12	anbi1d 685	. . . . . . . . . 10 |- (x = z -> ((xAy /\ zAw) <-> (zAy /\ zAw)))
14	13	imbi1d 308	. . . . . . . . 9 |- (x = z -> (((xAy /\ zAw) -> y = w) <-> ((zAy /\ zAw) -> y = w)))
15	11, 14	equsal 1960	. . . . . . . 8 |- (A.x(x = z -> ((xAy /\ zAw) -> y = w)) <-> ((zAy /\ zAw) -> y = w))
16	15	albii 1566	. . . . . . 7 |- (A.yA.x(x = z -> ((xAy /\ zAw) -> y = w)) <-> A.y((zAy /\ zAw) -> y = w))
17	10, 16	bitri 240	. . . . . 6 |- (A.xA.y(x = z -> ((xAy /\ zAw) -> y = w)) <-> A.y((zAy /\ zAw) -> y = w))
18		nfv 1619	. . . . . . . 8 |- F/y((xAw /\ zAw) -> x = z)
19		breq2 4644	. . . . . . . . . 10 |- (y = w -> (xAy <-> xAw))
20	19	anbi1d 685	. . . . . . . . 9 |- (y = w -> ((xAy /\ zAw) <-> (xAw /\ zAw)))
21	20	imbi1d 308	. . . . . . . 8 |- (y = w -> (((xAy /\ zAw) -> x = z) <-> ((xAw /\ zAw) -> x = z)))
22	18, 21	equsal 1960	. . . . . . 7 |- (A.y(y = w -> ((xAy /\ zAw) -> x = z)) <-> ((xAw /\ zAw) -> x = z))
23	22	albii 1566	. . . . . 6 |- (A.xA.y(y = w -> ((xAy /\ zAw) -> x = z)) <-> A.x((xAw /\ zAw) -> x = z))
24	17, 23	anbi12i 678	. . . . 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 262	. . . 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 1567	. . 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 1594	. . 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 241	. 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		dffun2 5120	. . . 4 |- (Fun A <-> A.zA.yA.w((zAy /\ zAw) -> y = w))
30		alcom 1737	. . . . 5 |- (A.yA.w((zAy /\ zAw) -> y = w) <-> A.wA.y((zAy /\ zAw) -> y = w))
31	30	albii 1566	. . . 4 |- (A.zA.yA.w((zAy /\ zAw) -> y = w) <-> A.zA.wA.y((zAy /\ zAw) -> y = w))
32	29, 31	bitri 240	. . 3 |- (Fun A <-> A.zA.wA.y((zAy /\ zAw) -> y = w))
33		brcnv 4893	. . . . . . . 8 |- (w`'Ax <-> xAw)
34		brcnv 4893	. . . . . . . 8 |- (w`'Az <-> zAw)
35	33, 34	anbi12i 678	. . . . . . 7 |- ((w`'Ax /\ w`'Az) <-> (xAw /\ zAw))
36	35	imbi1i 315	. . . . . 6 |- (((w`'Ax /\ w`'Az) -> x = z) <-> ((xAw /\ zAw) -> x = z))
37	36	albii 1566	. . . . 5 |- (A.z((w`'Ax /\ w`'Az) -> x = z) <-> A.z((xAw /\ zAw) -> x = z))
38	37	2albii 1567	. . . 4 |- (A.wA.xA.z((w`'Ax /\ w`'Az) -> x = z) <-> A.wA.xA.z((xAw /\ zAw) -> x = z))
39		dffun2 5120	. . . 4 |- (Fun `'A <-> A.wA.xA.z((w`'Ax /\ w`'Az) -> x = z))
40		alrot3 1738	. . . 4 |- (A.zA.wA.x((xAw /\ zAw) -> x = z) <-> A.wA.xA.z((xAw /\ zAw) -> x = z))
41	38, 39, 40	3bitr4i 268	. . 3 |- (Fun `'A <-> A.zA.wA.x((xAw /\ zAw) -> x = z))
42	32, 41	anbi12i 678	. 2 |- ((Fun A /\ Fun `'A) <-> (A.zA.wA.y((zAy /\ zAw) -> y = w) /\ A.zA.wA.x((xAw /\ zAw) -> x = z)))
43		alrot4 1739	. 2 |- (A.xA.yA.zA.w((xAy /\ zAw) -> (x = z <-> y = w)) <-> A.zA.wA.xA.y((xAy /\ zAw) -> (x = z <-> y = w)))
44	28, 42, 43	3bitr4i 268	1 |- ((Fun A /\ Fun `'A) <-> A.xA.yA.zA.w((xAy /\ zAw) -> (x = z <-> y = w)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   class class class wbr 4640  `'ccnv 4772  Fun wfun 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-id 4768  df-cnv 4786  df-fun 4790

This theorem is referenced by: (None)