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Theorem dff13 7178
Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)
Assertion
Ref Expression
dff13 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem dff13
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dff12 6714 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧))
2 ffn 6645 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 vex 3445 . . . . . . . . . . . . . . 15 𝑥 ∈ V
4 vex 3445 . . . . . . . . . . . . . . 15 𝑧 ∈ V
53, 4breldm 5844 . . . . . . . . . . . . . 14 (𝑥𝐹𝑧𝑥 ∈ dom 𝐹)
6 fndm 6582 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76eleq2d 2822 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑥 ∈ dom 𝐹𝑥𝐴))
85, 7syl5ib 243 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑥𝐹𝑧𝑥𝐴))
9 vex 3445 . . . . . . . . . . . . . . 15 𝑦 ∈ V
109, 4breldm 5844 . . . . . . . . . . . . . 14 (𝑦𝐹𝑧𝑦 ∈ dom 𝐹)
116eleq2d 2822 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹𝑦𝐴))
1210, 11syl5ib 243 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑦𝐹𝑧𝑦𝐴))
138, 12anim12d 609 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧𝑦𝐹𝑧) → (𝑥𝐴𝑦𝐴)))
1413pm4.71rd 563 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧𝑦𝐹𝑧) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝑥𝐹𝑧𝑦𝐹𝑧))))
15 eqcom 2743 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑧)
16 fnbrfvb 6872 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑧𝑥𝐹𝑧))
1715, 16bitrid 282 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑧 = (𝐹𝑥) ↔ 𝑥𝐹𝑧))
18 eqcom 2743 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑦) ↔ (𝐹𝑦) = 𝑧)
19 fnbrfvb 6872 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴𝑦𝐴) → ((𝐹𝑦) = 𝑧𝑦𝐹𝑧))
2018, 19bitrid 282 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝑦𝐴) → (𝑧 = (𝐹𝑦) ↔ 𝑦𝐹𝑧))
2117, 20bi2anan9 636 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴𝑥𝐴) ∧ (𝐹 Fn 𝐴𝑦𝐴)) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) ↔ (𝑥𝐹𝑧𝑦𝐹𝑧)))
2221anandis 675 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) ↔ (𝑥𝐹𝑧𝑦𝐹𝑧)))
2322pm5.32da 579 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → (((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝑥𝐹𝑧𝑦𝐹𝑧))))
2414, 23bitr4d 281 . . . . . . . . . 10 (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧𝑦𝐹𝑧) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)))))
2524imbi1d 341 . . . . . . . . 9 (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ (((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))) → 𝑥 = 𝑦)))
26 impexp 451 . . . . . . . . 9 ((((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)))
2725, 26bitrdi 286 . . . . . . . 8 (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))))
2827albidv 1922 . . . . . . 7 (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))))
29 19.21v 1941 . . . . . . . 8 (∀𝑧((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥𝐴𝑦𝐴) → ∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)))
30 19.23v 1944 . . . . . . . . . 10 (∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))
31 fvex 6832 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
3231eqvinc 3588 . . . . . . . . . . 11 ((𝐹𝑥) = (𝐹𝑦) ↔ ∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)))
3332imbi1i 349 . . . . . . . . . 10 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))
3430, 33bitr4i 277 . . . . . . . . 9 (∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3534imbi2i 335 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) → ∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3629, 35bitri 274 . . . . . . 7 (∀𝑧((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3728, 36bitrdi 286 . . . . . 6 (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
38372albidv 1925 . . . . 5 (𝐹 Fn 𝐴 → (∀𝑥𝑦𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
39 breq1 5092 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐹𝑧𝑦𝐹𝑧))
4039mo4 2564 . . . . . . 7 (∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝑦((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
4140albii 1820 . . . . . 6 (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑧𝑥𝑦((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
42 alrot3 2156 . . . . . 6 (∀𝑧𝑥𝑦((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
4341, 42bitri 274 . . . . 5 (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
44 r2al 3187 . . . . 5 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4538, 43, 443bitr4g 313 . . . 4 (𝐹 Fn 𝐴 → (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
462, 45syl 17 . . 3 (𝐹:𝐴𝐵 → (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4746pm5.32i 575 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
481, 47bitri 274 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1538   = wceq 1540  wex 1780  wcel 2105  ∃*wmo 2536  wral 3061   class class class wbr 5089  dom cdm 5614   Fn wfn 6468  wf 6469  1-1wf1 6470  cfv 6473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fv 6481
This theorem is referenced by:  dff13f  7179  f1veqaeq  7180  fpropnf1  7190  dff14a  7193  dff1o6  7197  fcof1  7209  nf1const  7226  soisoi  7249  f1opr  7385  f1o2ndf1  8022  fnwelem  8031  smo11  8257  tz7.48lem  8334  omsmo  8551  unxpdomlem3  9109  unfilem2  9168  fofinf1o  9184  inf3lem6  9482  r111  9624  fseqenlem1  9873  fodomacn  9905  alephf1  9934  alephiso  9947  ackbij1lem17  10085  infpssrlem5  10156  fin23lem28  10189  fin1a2lem2  10250  fin1a2lem4  10252  axcc2lem  10285  domtriomlem  10291  cnref1o  12818  injresinj  13601  om2uzf1oi  13766  cshf1  14613  wwlktovf1  14763  reeff1  15920  bitsf1  16244  crth  16568  eulerthlem2  16572  1arith  16717  vdwlem12  16782  xpsff1o  17367  setcmon  17891  fthestrcsetc  17956  embedsetcestrclem  17963  fthsetcestrc  17971  yoniso  18092  ghmf1  18951  orbsta  19007  symgextf1  19117  symgfixf1  19133  odf1  19257  kerf1ghm  20077  znf1o  20857  cygznlem3  20875  uvcf1  21097  lindff1  21125  mvrf1  21292  ply1sclf1  21558  scmatf1  21778  mdetunilem8  21866  mat2pmatf1  21976  pm2mpf1  22046  ist0-4  22978  ovolicc2lem4  24782  recosf1o  25789  efif1olem4  25799  basellem4  26331  dvdsmulf1o  26441  lgsqrlem2  26593  lgseisenlem2  26622  2lgslem1b  26638  axlowdimlem15  27526  upgrwlkdvdelem  28305  wlkswwlksf1o  28445  wwlksnextinj  28465  clwlkclwwlkf1  28575  clwwlkf1  28614  frgrncvvdeqlem8  28871  numclwwlk1lem2f1  28922  pjmf1  30279  unopf1o  30479  2ndresdju  31186  fnpreimac  31208  s3f1  31421  ccatf1  31423  swrdf1  31428  tocyccntz  31611  dff15  33268  f1resrcmplf1d  33271  f1resfz0f1d  33284  erdszelem9  33373  mrsubff1  33688  msubff1  33730  mvhf1  33733  f1omptsnlem  35605  fvineqsnf1  35679  fvineqsneu  35680  poimirlem26  35901  poimirlem27  35902  grpokerinj  36149  cdleme50f1  38804  dihf11  39528  sticksstones2  40353  dnnumch3  41123  wessf1ornlem  43038  projf1o  43052  sumnnodd  43496  dvnprodlem1  43812  fourierdlem34  44007  fourierdlem51  44023  fsetsnf1  44886  cfsetsnfsetf1  44893  fcoresf1  44903  imasetpreimafvbijlemf1  45196  fargshiftf1  45233  sprsymrelf1  45288  prproropf1o  45299  fmtnof1  45327  prmdvdsfmtnof1  45379  isomuspgrlem2c  45622  uspgrsprf1  45649  1arymaptf1  46328  2arymaptf1  46339  rrx2xpref1o  46404
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