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Theorem fun11 5159
 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) ↔ xyzw((xAy zAw) → (x = zy = w)))
Distinct variable group:   x,y,z,w,A

Proof of Theorem fun11
StepHypRef Expression
1 dfbi2 609 . . . . . . . 8 ((x = zy = w) ↔ ((x = zy = w) (y = wx = z)))
21imbi2i 303 . . . . . . 7 (((xAy zAw) → (x = zy = w)) ↔ ((xAy zAw) → ((x = zy = w) (y = wx = z))))
3 pm4.76 836 . . . . . . 7 ((((xAy zAw) → (x = zy = w)) ((xAy zAw) → (y = wx = z))) ↔ ((xAy zAw) → ((x = zy = w) (y = wx = z))))
4 bi2.04 350 . . . . . . . 8 (((xAy zAw) → (x = zy = w)) ↔ (x = z → ((xAy zAw) → y = w)))
5 bi2.04 350 . . . . . . . 8 (((xAy zAw) → (y = wx = z)) ↔ (y = w → ((xAy zAw) → x = z)))
64, 5anbi12i 678 . . . . . . 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 264 . . . . . 6 (((xAy zAw) → (x = zy = w)) ↔ ((x = z → ((xAy zAw) → y = w)) (y = w → ((xAy zAw) → x = z))))
872albii 1567 . . . . 5 (xy((xAy zAw) → (x = zy = w)) ↔ xy((x = z → ((xAy zAw) → y = w)) (y = w → ((xAy zAw) → x = z))))
9 19.26-2 1594 . . . . 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 1737 . . . . . . 7 (xy(x = z → ((xAy zAw) → y = w)) ↔ yx(x = z → ((xAy zAw) → y = w)))
11 nfv 1619 . . . . . . . . 9 x((zAy zAw) → y = w)
12 breq1 4642 . . . . . . . . . . 11 (x = z → (xAyzAy))
1312anbi1d 685 . . . . . . . . . 10 (x = z → ((xAy zAw) ↔ (zAy zAw)))
1413imbi1d 308 . . . . . . . . 9 (x = z → (((xAy zAw) → y = w) ↔ ((zAy zAw) → y = w)))
1511, 14equsal 1960 . . . . . . . 8 (x(x = z → ((xAy zAw) → y = w)) ↔ ((zAy zAw) → y = w))
1615albii 1566 . . . . . . 7 (yx(x = z → ((xAy zAw) → y = w)) ↔ y((zAy zAw) → y = w))
1710, 16bitri 240 . . . . . 6 (xy(x = z → ((xAy zAw) → y = w)) ↔ y((zAy zAw) → y = w))
18 nfv 1619 . . . . . . . 8 y((xAw zAw) → x = z)
19 breq2 4643 . . . . . . . . . 10 (y = w → (xAyxAw))
2019anbi1d 685 . . . . . . . . 9 (y = w → ((xAy zAw) ↔ (xAw zAw)))
2120imbi1d 308 . . . . . . . 8 (y = w → (((xAy zAw) → x = z) ↔ ((xAw zAw) → x = z)))
2218, 21equsal 1960 . . . . . . 7 (y(y = w → ((xAy zAw) → x = z)) ↔ ((xAw zAw) → x = z))
2322albii 1566 . . . . . 6 (xy(y = w → ((xAy zAw) → x = z)) ↔ x((xAw zAw) → x = z))
2417, 23anbi12i 678 . . . . 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 262 . . . 4 (xy((xAy zAw) → (x = zy = w)) ↔ (y((zAy zAw) → y = w) x((xAw zAw) → x = z)))
26252albii 1567 . . 3 (zwxy((xAy zAw) → (x = zy = w)) ↔ zw(y((zAy zAw) → y = w) x((xAw zAw) → x = z)))
27 19.26-2 1594 . . 3 (zw(y((zAy zAw) → y = w) x((xAw zAw) → x = z)) ↔ (zwy((zAy zAw) → y = w) zwx((xAw zAw) → x = z)))
2826, 27bitr2i 241 . 2 ((zwy((zAy zAw) → y = w) zwx((xAw zAw) → x = z)) ↔ zwxy((xAy zAw) → (x = zy = w)))
29 dffun2 5119 . . . 4 (Fun Azyw((zAy zAw) → y = w))
30 alcom 1737 . . . . 5 (yw((zAy zAw) → y = w) ↔ wy((zAy zAw) → y = w))
3130albii 1566 . . . 4 (zyw((zAy zAw) → y = w) ↔ zwy((zAy zAw) → y = w))
3229, 31bitri 240 . . 3 (Fun Azwy((zAy zAw) → y = w))
33 brcnv 4892 . . . . . . . 8 (wAxxAw)
34 brcnv 4892 . . . . . . . 8 (wAzzAw)
3533, 34anbi12i 678 . . . . . . 7 ((wAx wAz) ↔ (xAw zAw))
3635imbi1i 315 . . . . . 6 (((wAx wAz) → x = z) ↔ ((xAw zAw) → x = z))
3736albii 1566 . . . . 5 (z((wAx wAz) → x = z) ↔ z((xAw zAw) → x = z))
38372albii 1567 . . . 4 (wxz((wAx wAz) → x = z) ↔ wxz((xAw zAw) → x = z))
39 dffun2 5119 . . . 4 (Fun Awxz((wAx wAz) → x = z))
40 alrot3 1738 . . . 4 (zwx((xAw zAw) → x = z) ↔ wxz((xAw zAw) → x = z))
4138, 39, 403bitr4i 268 . . 3 (Fun Azwx((xAw zAw) → x = z))
4232, 41anbi12i 678 . 2 ((Fun A Fun A) ↔ (zwy((zAy zAw) → y = w) zwx((xAw zAw) → x = z)))
43 alrot4 1739 . 2 (xyzw((xAy zAw) → (x = zy = w)) ↔ zwxy((xAy zAw) → (x = zy = w)))
4428, 42, 433bitr4i 268 1 ((Fun A Fun A) ↔ xyzw((xAy zAw) → (x = zy = w)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   class class class wbr 4639  ◡ccnv 4771  Fun wfun 4775 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-id 4767  df-cnv 4785  df-fun 4789 This theorem is referenced by: (None)
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