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

Theorem dffun2 6563
Description: Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2129, ax-12 2166. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2146. (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 6555 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 cotrg 6118 . . . 4 ((𝐴𝐴) ⊆ I ↔ ∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
3 breq1 5155 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝐴𝑥𝑤𝐴𝑥))
43anbi1d 629 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑤𝐴𝑥𝑥𝐴𝑧)))
5 breq1 5155 . . . . . . . 8 (𝑦 = 𝑤 → (𝑦 I 𝑧𝑤 I 𝑧))
64, 5imbi12d 343 . . . . . . 7 (𝑦 = 𝑤 → (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑤𝐴𝑥𝑥𝐴𝑧) → 𝑤 I 𝑧)))
76albidv 1915 . . . . . 6 (𝑦 = 𝑤 → (∀𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑤𝐴𝑥𝑥𝐴𝑧) → 𝑤 I 𝑧)))
8 breq2 5156 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑦𝐴𝑥𝑦𝐴𝑤))
9 breq1 5155 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥𝐴𝑧𝑤𝐴𝑧))
108, 9anbi12d 630 . . . . . . . 8 (𝑥 = 𝑤 → ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑦𝐴𝑤𝑤𝐴𝑧)))
1110imbi1d 340 . . . . . . 7 (𝑥 = 𝑤 → (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑦𝐴𝑤𝑤𝐴𝑧) → 𝑦 I 𝑧)))
1211albidv 1915 . . . . . 6 (𝑥 = 𝑤 → (∀𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑦𝐴𝑤𝑤𝐴𝑧) → 𝑦 I 𝑧)))
137, 12alcomw 2039 . . . . 5 (∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
14 vex 3477 . . . . . . . . 9 𝑦 ∈ V
15 vex 3477 . . . . . . . . 9 𝑥 ∈ V
1614, 15brcnv 5889 . . . . . . . 8 (𝑦𝐴𝑥𝑥𝐴𝑦)
1716anbi1i 622 . . . . . . 7 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
18 vex 3477 . . . . . . . 8 𝑧 ∈ V
1918ideq 5859 . . . . . . 7 (𝑦 I 𝑧𝑦 = 𝑧)
2017, 19imbi12i 349 . . . . . 6 (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
21203albii 1815 . . . . 5 (∀𝑥𝑦𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2213, 21bitri 274 . . . 4 (∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
232, 22bitri 274 . . 3 ((𝐴𝐴) ⊆ I ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2423anbi2i 621 . 2 ((Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
251, 24bitri 274 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wss 3949   class class class wbr 5152   I cid 5579  ccnv 5681  ccom 5686  Rel wrel 5687  Fun wfun 6547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-fun 6555
This theorem is referenced by:  dffun6  6566  dffun3OLD  6568  dffun4  6569  fundif  6607  fliftfun  7326  frrlem9  8306  fprlem1  8312  wfrlem5OLD  8340  wfrfunOLD  8346  frrlem15  9788  fpwwe2lem10  10671  fclim  15537  invfun  17754  lmfun  23305  ulmdm  26349  fundmpss  35395  fununiq  35397  fnsingle  35548  funimage  35557  funpartfun  35572
  Copyright terms: Public domain W3C validator