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Theorem dff13 6552
 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 6138 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧))
2 ffn 6083 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 vex 3234 . . . . . . . . . . . . . . 15 𝑥 ∈ V
4 vex 3234 . . . . . . . . . . . . . . 15 𝑧 ∈ V
53, 4breldm 5361 . . . . . . . . . . . . . 14 (𝑥𝐹𝑧𝑥 ∈ dom 𝐹)
6 fndm 6028 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76eleq2d 2716 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑥 ∈ dom 𝐹𝑥𝐴))
85, 7syl5ib 234 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑥𝐹𝑧𝑥𝐴))
9 vex 3234 . . . . . . . . . . . . . . 15 𝑦 ∈ V
109, 4breldm 5361 . . . . . . . . . . . . . 14 (𝑦𝐹𝑧𝑦 ∈ dom 𝐹)
116eleq2d 2716 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹𝑦𝐴))
1210, 11syl5ib 234 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑦𝐹𝑧𝑦𝐴))
138, 12anim12d 585 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧𝑦𝐹𝑧) → (𝑥𝐴𝑦𝐴)))
1413pm4.71rd 668 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧𝑦𝐹𝑧) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝑥𝐹𝑧𝑦𝐹𝑧))))
15 eqcom 2658 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑧)
16 fnbrfvb 6274 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑧𝑥𝐹𝑧))
1715, 16syl5bb 272 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑧 = (𝐹𝑥) ↔ 𝑥𝐹𝑧))
18 eqcom 2658 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑦) ↔ (𝐹𝑦) = 𝑧)
19 fnbrfvb 6274 . . . . . . . . . . . . . . 15 ((𝐹 Fn 𝐴𝑦𝐴) → ((𝐹𝑦) = 𝑧𝑦𝐹𝑧))
2018, 19syl5bb 272 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝑦𝐴) → (𝑧 = (𝐹𝑦) ↔ 𝑦𝐹𝑧))
2117, 20bi2anan9 935 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴𝑥𝐴) ∧ (𝐹 Fn 𝐴𝑦𝐴)) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) ↔ (𝑥𝐹𝑧𝑦𝐹𝑧)))
2221anandis 890 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) ↔ (𝑥𝐹𝑧𝑦𝐹𝑧)))
2322pm5.32da 674 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → (((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝑥𝐹𝑧𝑦𝐹𝑧))))
2414, 23bitr4d 271 . . . . . . . . . 10 (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧𝑦𝐹𝑧) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)))))
2524imbi1d 330 . . . . . . . . 9 (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ (((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))) → 𝑥 = 𝑦)))
26 impexp 461 . . . . . . . . 9 ((((𝑥𝐴𝑦𝐴) ∧ (𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦))) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)))
2725, 26syl6bb 276 . . . . . . . 8 (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))))
2827albidv 1889 . . . . . . 7 (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))))
29 19.21v 1908 . . . . . . . 8 (∀𝑧((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥𝐴𝑦𝐴) → ∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)))
30 19.23v 1911 . . . . . . . . . 10 (∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))
31 fvex 6239 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
3231eqvinc 3361 . . . . . . . . . . 11 ((𝐹𝑥) = (𝐹𝑦) ↔ ∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)))
3332imbi1i 338 . . . . . . . . . 10 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦))
3430, 33bitr4i 267 . . . . . . . . 9 (∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3534imbi2i 325 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) → ∀𝑧((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3629, 35bitri 264 . . . . . . 7 (∀𝑧((𝑥𝐴𝑦𝐴) → ((𝑧 = (𝐹𝑥) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3728, 36syl6bb 276 . . . . . 6 (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
38372albidv 1891 . . . . 5 (𝐹 Fn 𝐴 → (∀𝑥𝑦𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
39 breq1 4688 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐹𝑧𝑦𝐹𝑧))
4039mo4 2546 . . . . . . 7 (∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝑦((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
4140albii 1787 . . . . . 6 (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑧𝑥𝑦((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
42 alrot3 2078 . . . . . 6 (∀𝑧𝑥𝑦((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
4341, 42bitri 264 . . . . 5 (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑧𝑦𝐹𝑧) → 𝑥 = 𝑦))
44 r2al 2968 . . . . 5 (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4538, 43, 443bitr4g 303 . . . 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 670 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
481, 47bitri 264 1 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃*wmo 2499  ∀wral 2941   class class class wbr 4685  dom cdm 5143   Fn wfn 5921  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934 This theorem is referenced by:  dff13f  6553  f1veqaeq  6554  fpropnf1  6564  dff14a  6567  dff1o6  6571  fcof1  6582  soisoi  6618  f1o2ndf1  7330  fnwelem  7337  smo11  7506  tz7.48lem  7581  omsmo  7779  unxpdomlem3  8207  unfilem2  8266  fofinf1o  8282  inf3lem6  8568  r111  8676  fseqenlem1  8885  fodomacn  8917  alephf1  8946  alephiso  8959  ackbij1lem17  9096  infpssrlem5  9167  fin23lem28  9200  fin1a2lem2  9261  fin1a2lem4  9263  axcc2lem  9296  domtriomlem  9302  cnref1o  11865  injresinj  12629  om2uzf1oi  12792  cshf1  13602  wwlktovf1  13746  reeff1  14894  bitsf1  15215  crth  15530  eulerthlem2  15534  1arith  15678  vdwlem12  15743  xpsff1o  16275  setcmon  16784  fthestrcsetc  16837  embedsetcestrclem  16844  fthsetcestrc  16852  yoniso  16972  ghmf1  17736  orbsta  17792  symgextf1  17887  symgfixf1  17903  odf1  18025  kerf1hrm  18791  mvrf1  19473  ply1sclf1  19707  znf1o  19948  cygznlem3  19966  uvcf1  20179  lindff1  20207  scmatf1  20385  mdetunilem8  20473  mat2pmatf1  20582  pm2mpf1  20652  ist0-4  21580  ovolicc2lem4  23334  recosf1o  24326  efif1olem4  24336  basellem4  24855  dvdsmulf1o  24965  lgsqrlem2  25117  lgseisenlem2  25146  2lgslem1b  25162  axlowdimlem15  25881  upgrwlkdvdelem  26688  wlkpwwlkf1ouspgr  26833  wlknwwlksninj  26843  wlkwwlkinj  26850  wwlksnextinj  26862  clwwlkf1  27012  clwlksf1clwwlk  27056  frgrncvvdeqlem8  27286  numclwlk1lem2f1  27347  pjmf1  28703  unopf1o  28903  erdszelem9  31307  mrsubff1  31537  msubff1  31579  mvhf1  31582  f1omptsnlem  33313  poimirlem26  33565  poimirlem27  33566  f1opr  33649  grpokerinj  33822  cdleme50f1  36148  dihf11  36873  dnnumch3  37934  wessf1ornlem  39685  projf1o  39700  sumnnodd  40180  dvnprodlem1  40479  fourierdlem34  40676  fourierdlem51  40692  fargshiftf1  41702  fmtnof1  41772  prmdvdsfmtnof1  41824  sprsymrelf1  42071  uspgrsprf1  42080
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