ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dffun2 GIF version

Theorem dffun2 5012
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 5004 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 df-id 4111 . . . . . 6 I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
32sseq2i 3049 . . . . 5 ((𝐴𝐴) ⊆ I ↔ (𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
4 df-co 4437 . . . . . 6 (𝐴𝐴) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)}
54sseq1i 3048 . . . . 5 ((𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
6 ssopab2b 4094 . . . . 5 ({⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ ∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧))
73, 5, 63bitri 204 . . . 4 ((𝐴𝐴) ⊆ I ↔ ∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧))
8 vex 2622 . . . . . . . . . . . 12 𝑦 ∈ V
9 vex 2622 . . . . . . . . . . . 12 𝑥 ∈ V
108, 9brcnv 4607 . . . . . . . . . . 11 (𝑦𝐴𝑥𝑥𝐴𝑦)
1110anbi1i 446 . . . . . . . . . 10 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
1211exbii 1541 . . . . . . . . 9 (∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) ↔ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧))
1312imbi1i 236 . . . . . . . 8 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
14 19.23v 1811 . . . . . . . 8 (∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1513, 14bitr4i 185 . . . . . . 7 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1615albii 1404 . . . . . 6 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
17 alcom 1412 . . . . . 6 (∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1816, 17bitri 182 . . . . 5 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1918albii 1404 . . . 4 (∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
20 alcom 1412 . . . 4 (∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
217, 19, 203bitri 204 . . 3 ((𝐴𝐴) ⊆ I ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2221anbi2i 445 . 2 ((Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
231, 22bitri 182 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287  wex 1426  wss 2997   class class class wbr 3837  {copab 3890   I cid 4106  ccnv 4427  ccom 4432  Rel wrel 4433  Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-cnv 4436  df-co 4437  df-fun 5004
This theorem is referenced by:  dffun4  5013  dffun6f  5015  sbcfung  5025  funcnveq  5063  fliftfun  5557  fclim  10646
  Copyright terms: Public domain W3C validator