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Theorem fun11 4909
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). (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
fun11 ((Fun A Fun A) ↔ xyzw((xAy zAw) → (x = zy = w)))
Distinct variable group:   x,y,z,w,A

Proof of Theorem fun11
StepHypRef Expression
1 dfbi2 368 . . . . . . . 8 ((x = zy = w) ↔ ((x = zy = w) (y = wx = z)))
21imbi2i 215 . . . . . . 7 (((xAy zAw) → (x = zy = w)) ↔ ((xAy zAw) → ((x = zy = w) (y = wx = z))))
3 pm4.76 536 . . . . . . 7 ((((xAy zAw) → (x = zy = w)) ((xAy zAw) → (y = wx = z))) ↔ ((xAy zAw) → ((x = zy = w) (y = wx = z))))
4 bi2.04 237 . . . . . . . 8 (((xAy zAw) → (x = zy = w)) ↔ (x = z → ((xAy zAw) → y = w)))
5 bi2.04 237 . . . . . . . 8 (((xAy zAw) → (y = wx = z)) ↔ (y = w → ((xAy zAw) → x = z)))
64, 5anbi12i 433 . . . . . . 7 ((((xAy zAw) → (x = zy = w)) ((xAy zAw) → (y = wx = z))) ↔ ((x = z → ((xAy zAw) → y = w)) (y = w → ((xAy zAw) → x = z))))
72, 3, 63bitr2i 197 . . . . . 6 (((xAy zAw) → (x = zy = w)) ↔ ((x = z → ((xAy zAw) → y = w)) (y = w → ((xAy zAw) → x = z))))
872albii 1357 . . . . 5 (xy((xAy zAw) → (x = zy = w)) ↔ xy((x = z → ((xAy zAw) → y = w)) (y = w → ((xAy zAw) → x = z))))
9 19.26-2 1368 . . . . 5 (xy((x = z → ((xAy zAw) → y = w)) (y = w → ((xAy zAw) → x = z))) ↔ (xy(x = z → ((xAy zAw) → y = w)) xy(y = w → ((xAy zAw) → x = z))))
10 alcom 1364 . . . . . . 7 (xy(x = z → ((xAy zAw) → y = w)) ↔ yx(x = z → ((xAy zAw) → y = w)))
11 nfv 1418 . . . . . . . . 9 x((zAy zAw) → y = w)
12 breq1 3758 . . . . . . . . . . 11 (x = z → (xAyzAy))
1312anbi1d 438 . . . . . . . . . 10 (x = z → ((xAy zAw) ↔ (zAy zAw)))
1413imbi1d 220 . . . . . . . . 9 (x = z → (((xAy zAw) → y = w) ↔ ((zAy zAw) → y = w)))
1511, 14equsal 1612 . . . . . . . 8 (x(x = z → ((xAy zAw) → y = w)) ↔ ((zAy zAw) → y = w))
1615albii 1356 . . . . . . 7 (yx(x = z → ((xAy zAw) → y = w)) ↔ y((zAy zAw) → y = w))
1710, 16bitri 173 . . . . . 6 (xy(x = z → ((xAy zAw) → y = w)) ↔ y((zAy zAw) → y = w))
18 nfv 1418 . . . . . . . 8 y((xAw zAw) → x = z)
19 breq2 3759 . . . . . . . . . 10 (y = w → (xAyxAw))
2019anbi1d 438 . . . . . . . . 9 (y = w → ((xAy zAw) ↔ (xAw zAw)))
2120imbi1d 220 . . . . . . . 8 (y = w → (((xAy zAw) → x = z) ↔ ((xAw zAw) → x = z)))
2218, 21equsal 1612 . . . . . . 7 (y(y = w → ((xAy zAw) → x = z)) ↔ ((xAw zAw) → x = z))
2322albii 1356 . . . . . 6 (xy(y = w → ((xAy zAw) → x = z)) ↔ x((xAw zAw) → x = z))
2417, 23anbi12i 433 . . . . 5 ((xy(x = z → ((xAy zAw) → y = w)) xy(y = w → ((xAy zAw) → x = z))) ↔ (y((zAy zAw) → y = w) x((xAw zAw) → x = z)))
258, 9, 243bitri 195 . . . 4 (xy((xAy zAw) → (x = zy = w)) ↔ (y((zAy zAw) → y = w) x((xAw zAw) → x = z)))
26252albii 1357 . . 3 (zwxy((xAy zAw) → (x = zy = w)) ↔ zw(y((zAy zAw) → y = w) x((xAw zAw) → x = z)))
27 19.26-2 1368 . . 3 (zw(y((zAy zAw) → y = w) x((xAw zAw) → x = z)) ↔ (zwy((zAy zAw) → y = w) zwx((xAw zAw) → x = z)))
2826, 27bitr2i 174 . 2 ((zwy((zAy zAw) → y = w) zwx((xAw zAw) → x = z)) ↔ zwxy((xAy zAw) → (x = zy = w)))
29 fun2cnv 4906 . . . 4 (Fun Az∃*y zAy)
30 breq2 3759 . . . . . 6 (y = w → (zAyzAw))
3130mo4 1958 . . . . 5 (∃*y zAyyw((zAy zAw) → y = w))
3231albii 1356 . . . 4 (z∃*y zAyzyw((zAy zAw) → y = w))
33 alcom 1364 . . . . 5 (yw((zAy zAw) → y = w) ↔ wy((zAy zAw) → y = w))
3433albii 1356 . . . 4 (zyw((zAy zAw) → y = w) ↔ zwy((zAy zAw) → y = w))
3529, 32, 343bitri 195 . . 3 (Fun Azwy((zAy zAw) → y = w))
36 funcnv2 4902 . . . 4 (Fun Aw∃*x xAw)
37 breq1 3758 . . . . . 6 (x = z → (xAwzAw))
3837mo4 1958 . . . . 5 (∃*x xAwxz((xAw zAw) → x = z))
3938albii 1356 . . . 4 (w∃*x xAwwxz((xAw zAw) → x = z))
40 alcom 1364 . . . . . 6 (xz((xAw zAw) → x = z) ↔ zx((xAw zAw) → x = z))
4140albii 1356 . . . . 5 (wxz((xAw zAw) → x = z) ↔ wzx((xAw zAw) → x = z))
42 alcom 1364 . . . . 5 (wzx((xAw zAw) → x = z) ↔ zwx((xAw zAw) → x = z))
4341, 42bitri 173 . . . 4 (wxz((xAw zAw) → x = z) ↔ zwx((xAw zAw) → x = z))
4436, 39, 433bitri 195 . . 3 (Fun Azwx((xAw zAw) → x = z))
4535, 44anbi12i 433 . 2 ((Fun A Fun A) ↔ (zwy((zAy zAw) → y = w) zwx((xAw zAw) → x = z)))
46 alrot4 1372 . 2 (xyzw((xAy zAw) → (x = zy = w)) ↔ zwxy((xAy zAw) → (x = zy = w)))
4728, 45, 463bitr4i 201 1 ((Fun A Fun A) ↔ xyzw((xAy zAw) → (x = zy = w)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  ∃*wmo 1898   class class class wbr 3755  ccnv 4287  Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847
This theorem is referenced by: (None)
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