MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffun2 Structured version   Visualization version   GIF version

Theorem dffun2 6573
Description: Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2139, ax-12 2175. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2155. (Revised by BTernaryTau, 29-Dec-2024.)
Assertion
Ref Expression
dffun2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dffun2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-fun 6565 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 cotrg 6130 . . . 4 ((𝐴𝐴) ⊆ I ↔ ∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
3 breq1 5151 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝐴𝑥𝑤𝐴𝑥))
43anbi1d 631 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑤𝐴𝑥𝑥𝐴𝑧)))
5 breq1 5151 . . . . . . . 8 (𝑦 = 𝑤 → (𝑦 I 𝑧𝑤 I 𝑧))
64, 5imbi12d 344 . . . . . . 7 (𝑦 = 𝑤 → (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑤𝐴𝑥𝑥𝐴𝑧) → 𝑤 I 𝑧)))
76albidv 1918 . . . . . 6 (𝑦 = 𝑤 → (∀𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑤𝐴𝑥𝑥𝐴𝑧) → 𝑤 I 𝑧)))
8 breq2 5152 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑦𝐴𝑥𝑦𝐴𝑤))
9 breq1 5151 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥𝐴𝑧𝑤𝐴𝑧))
108, 9anbi12d 632 . . . . . . . 8 (𝑥 = 𝑤 → ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑦𝐴𝑤𝑤𝐴𝑧)))
1110imbi1d 341 . . . . . . 7 (𝑥 = 𝑤 → (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑦𝐴𝑤𝑤𝐴𝑧) → 𝑦 I 𝑧)))
1211albidv 1918 . . . . . 6 (𝑥 = 𝑤 → (∀𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑦𝐴𝑤𝑤𝐴𝑧) → 𝑦 I 𝑧)))
137, 12alcomw 2042 . . . . 5 (∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
14 vex 3482 . . . . . . . . 9 𝑦 ∈ V
15 vex 3482 . . . . . . . . 9 𝑥 ∈ V
1614, 15brcnv 5896 . . . . . . . 8 (𝑦𝐴𝑥𝑥𝐴𝑦)
1716anbi1i 624 . . . . . . 7 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
18 vex 3482 . . . . . . . 8 𝑧 ∈ V
1918ideq 5866 . . . . . . 7 (𝑦 I 𝑧𝑦 = 𝑧)
2017, 19imbi12i 350 . . . . . 6 (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
21203albii 1818 . . . . 5 (∀𝑥𝑦𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2213, 21bitri 275 . . . 4 (∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
232, 22bitri 275 . . 3 ((𝐴𝐴) ⊆ I ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2423anbi2i 623 . 2 ((Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
251, 24bitri 275 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wss 3963   class class class wbr 5148   I cid 5582  ccnv 5688  ccom 5693  Rel wrel 5694  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565
This theorem is referenced by:  dffun6  6576  dffun3OLD  6578  dffun4  6579  fundif  6617  fliftfun  7332  frrlem9  8318  fprlem1  8324  wfrlem5OLD  8352  wfrfunOLD  8358  frrlem15  9795  fpwwe2lem10  10678  fclim  15586  invfun  17812  lmfun  23405  ulmdm  26451  fundmpss  35748  fununiq  35750  fnsingle  35901  funimage  35910  funpartfun  35925
  Copyright terms: Public domain W3C validator