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Theorem dffun2 6133
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 6125 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 df-id 5250 . . . . . 6 I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
32sseq2i 3855 . . . . 5 ((𝐴𝐴) ⊆ I ↔ (𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
4 df-co 5351 . . . . . 6 (𝐴𝐴) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)}
54sseq1i 3854 . . . . 5 ((𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
6 ssopab2b 5228 . . . . 5 ({⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ ∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧))
73, 5, 63bitri 289 . . . 4 ((𝐴𝐴) ⊆ I ↔ ∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧))
8 vex 3417 . . . . . . . . . . . 12 𝑦 ∈ V
9 vex 3417 . . . . . . . . . . . 12 𝑥 ∈ V
108, 9brcnv 5537 . . . . . . . . . . 11 (𝑦𝐴𝑥𝑥𝐴𝑦)
1110anbi1i 619 . . . . . . . . . 10 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
1211exbii 1949 . . . . . . . . 9 (∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) ↔ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧))
1312imbi1i 341 . . . . . . . 8 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
14 19.23v 2043 . . . . . . . 8 (∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1513, 14bitr4i 270 . . . . . . 7 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1615albii 1920 . . . . . 6 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
17 alcom 2211 . . . . . 6 (∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1816, 17bitri 267 . . . . 5 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1918albii 1920 . . . 4 (∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
20 alcom 2211 . . . 4 (∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
217, 19, 203bitri 289 . . 3 ((𝐴𝐴) ⊆ I ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2221anbi2i 618 . 2 ((Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
231, 22bitri 267 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1656  wex 1880  wss 3798   class class class wbr 4873  {copab 4935   I cid 5249  ccnv 5341  ccom 5346  Rel wrel 5347  Fun wfun 6117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-cnv 5350  df-co 5351  df-fun 6125
This theorem is referenced by:  dffun3  6134  dffun4  6135  fundif  6171  fliftfun  6817  wfrlem5  7685  wfrfun  7691  fpwwe2lem11  9777  fclim  14661  invfun  16776  lmfun  21556  ulmdm  24546  fundmpss  32206  fununiq  32209  frrlem5  32323  frrlem5c  32325  fnsingle  32565  funimage  32574  funpartfun  32589
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