MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffun2OLD Structured version   Visualization version   GIF version

Theorem dffun2OLD 6548
Description: Obsolete version of dffun2 6547 as of 29-Dec-2024. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2129, ax-12 2163. (Revised by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dffun2OLD (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dffun2OLD
StepHypRef Expression
1 df-fun 6539 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 cotrg 6102 . . . 4 ((𝐴𝐴) ⊆ I ↔ ∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
3 alcom 2148 . . . . 5 (∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
4 vex 3472 . . . . . . . . 9 𝑦 ∈ V
5 vex 3472 . . . . . . . . 9 𝑥 ∈ V
64, 5brcnv 5876 . . . . . . . 8 (𝑦𝐴𝑥𝑥𝐴𝑦)
76anbi1i 623 . . . . . . 7 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
8 vex 3472 . . . . . . . 8 𝑧 ∈ V
98ideq 5846 . . . . . . 7 (𝑦 I 𝑧𝑦 = 𝑧)
107, 9imbi12i 350 . . . . . 6 (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
11103albii 1815 . . . . 5 (∀𝑥𝑦𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
123, 11bitri 275 . . . 4 (∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
132, 12bitri 275 . . 3 ((𝐴𝐴) ⊆ I ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1413anbi2i 622 . 2 ((Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
151, 14bitri 275 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531  wss 3943   class class class wbr 5141   I cid 5566  ccnv 5668  ccom 5673  Rel wrel 5674  Fun wfun 6531
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-11 2146  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-fun 6539
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator