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Theorem dffun2 5886
 Description: Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dffun2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem dffun2
StepHypRef Expression
1 df-fun 5878 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 df-id 5014 . . . . . 6 I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
32sseq2i 3622 . . . . 5 ((𝐴𝐴) ⊆ I ↔ (𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
4 df-co 5113 . . . . . 6 (𝐴𝐴) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)}
54sseq1i 3621 . . . . 5 ((𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
6 ssopab2b 4992 . . . . 5 ({⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ ∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧))
73, 5, 63bitri 286 . . . 4 ((𝐴𝐴) ⊆ I ↔ ∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧))
8 vex 3198 . . . . . . . . . . . 12 𝑦 ∈ V
9 vex 3198 . . . . . . . . . . . 12 𝑥 ∈ V
108, 9brcnv 5294 . . . . . . . . . . 11 (𝑦𝐴𝑥𝑥𝐴𝑦)
1110anbi1i 730 . . . . . . . . . 10 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
1211exbii 1772 . . . . . . . . 9 (∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) ↔ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧))
1312imbi1i 339 . . . . . . . 8 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
14 19.23v 1900 . . . . . . . 8 (∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1513, 14bitr4i 267 . . . . . . 7 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1615albii 1745 . . . . . 6 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
17 alcom 2035 . . . . . 6 (∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1816, 17bitri 264 . . . . 5 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1918albii 1745 . . . 4 (∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
20 alcom 2035 . . . 4 (∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
217, 19, 203bitri 286 . . 3 ((𝐴𝐴) ⊆ I ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2221anbi2i 729 . 2 ((Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
231, 22bitri 264 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1479  ∃wex 1702   ⊆ wss 3567   class class class wbr 4644  {copab 4703   I cid 5013  ◡ccnv 5103   ∘ ccom 5108  Rel wrel 5109  Fun wfun 5870 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-cnv 5112  df-co 5113  df-fun 5878 This theorem is referenced by:  dffun3  5887  dffun4  5888  fundif  5923  fliftfun  6547  wfrlem5  7404  wfrfun  7410  fpwwe2lem11  9447  fclim  14265  invfun  16405  lmfun  21166  ulmdm  24128  fundmpss  31640  fununiq  31643  frrlem5  31758  frrlem5c  31760  fnsingle  32001  funimage  32010  funpartfun  32025
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