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Theorem dffun2 6041
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 6033 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 df-id 5157 . . . . . 6 I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
32sseq2i 3779 . . . . 5 ((𝐴𝐴) ⊆ I ↔ (𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
4 df-co 5258 . . . . . 6 (𝐴𝐴) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)}
54sseq1i 3778 . . . . 5 ((𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
6 ssopab2b 5135 . . . . 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 3354 . . . . . . . . . . . 12 𝑦 ∈ V
9 vex 3354 . . . . . . . . . . . 12 𝑥 ∈ V
108, 9brcnv 5443 . . . . . . . . . . 11 (𝑦𝐴𝑥𝑥𝐴𝑦)
1110anbi1i 610 . . . . . . . . . 10 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
1211exbii 1924 . . . . . . . . 9 (∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) ↔ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧))
1312imbi1i 338 . . . . . . . 8 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
14 19.23v 2023 . . . . . . . 8 (∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1513, 14bitr4i 267 . . . . . . 7 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1615albii 1895 . . . . . 6 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
17 alcom 2193 . . . . . 6 (∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1816, 17bitri 264 . . . . 5 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1918albii 1895 . . . 4 (∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
20 alcom 2193 . . . 4 (∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
217, 19, 203bitri 286 . . 3 ((𝐴𝐴) ⊆ I ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2221anbi2i 609 . 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 382  wal 1629  wex 1852  wss 3723   class class class wbr 4786  {copab 4846   I cid 5156  ccnv 5248  ccom 5253  Rel wrel 5254  Fun wfun 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-cnv 5257  df-co 5258  df-fun 6033
This theorem is referenced by:  dffun3  6042  dffun4  6043  fundif  6078  fliftfun  6705  wfrlem5  7572  wfrfun  7578  fpwwe2lem11  9664  fclim  14492  invfun  16631  lmfun  21406  ulmdm  24367  fundmpss  32002  fununiq  32005  frrlem5  32121  frrlem5c  32123  fnsingle  32363  funimage  32372  funpartfun  32387
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