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Theorem dffun2 6570
Description: Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2140, ax-12 2176. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2156. (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 6562 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 cotrg 6126 . . . 4 ((𝐴𝐴) ⊆ I ↔ ∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
3 breq1 5145 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝐴𝑥𝑤𝐴𝑥))
43anbi1d 631 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑤𝐴𝑥𝑥𝐴𝑧)))
5 breq1 5145 . . . . . . . 8 (𝑦 = 𝑤 → (𝑦 I 𝑧𝑤 I 𝑧))
64, 5imbi12d 344 . . . . . . 7 (𝑦 = 𝑤 → (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑤𝐴𝑥𝑥𝐴𝑧) → 𝑤 I 𝑧)))
76albidv 1919 . . . . . 6 (𝑦 = 𝑤 → (∀𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑤𝐴𝑥𝑥𝐴𝑧) → 𝑤 I 𝑧)))
8 breq2 5146 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑦𝐴𝑥𝑦𝐴𝑤))
9 breq1 5145 . . . . . . . . 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 1919 . . . . . 6 (𝑥 = 𝑤 → (∀𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑧((𝑦𝐴𝑤𝑤𝐴𝑧) → 𝑦 I 𝑧)))
137, 12alcomw 2043 . . . . 5 (∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
14 vex 3483 . . . . . . . . 9 𝑦 ∈ V
15 vex 3483 . . . . . . . . 9 𝑥 ∈ V
1614, 15brcnv 5892 . . . . . . . 8 (𝑦𝐴𝑥𝑥𝐴𝑦)
1716anbi1i 624 . . . . . . 7 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
18 vex 3483 . . . . . . . 8 𝑧 ∈ V
1918ideq 5862 . . . . . . 7 (𝑦 I 𝑧𝑦 = 𝑧)
2017, 19imbi12i 350 . . . . . 6 (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
21203albii 1820 . . . . 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 1537  wss 3950   class class class wbr 5142   I cid 5576  ccnv 5683  ccom 5688  Rel wrel 5689  Fun wfun 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-fun 6562
This theorem is referenced by:  dffun6  6573  dffun3OLD  6575  dffun4  6576  fundif  6614  fliftfun  7333  frrlem9  8320  fprlem1  8326  wfrlem5OLD  8354  wfrfunOLD  8360  frrlem15  9798  fpwwe2lem10  10681  fclim  15590  invfun  17809  lmfun  23390  ulmdm  26437  fundmpss  35768  fununiq  35770  fnsingle  35921  funimage  35930  funpartfun  35945  functhincfun  49123
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