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Theorem dffun3 5644
Description: Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dffun3 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem dffun3
StepHypRef Expression
1 dffun2 5643 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
2 breq2 4438 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝐴𝑦𝑥𝐴𝑧))
32mo4 2400 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
4 mo2v 2360 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
53, 4bitr3i 261 . . . 4 (∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
65albii 1720 . . 3 (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
76anbi2i 717 . 2 ((Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
81, 7bitri 259 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  wa 378  wal 1466  wex 1692  ∃*wmo 2354   class class class wbr 4434  Rel wrel 4885  Fun wfun 5627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pr 4680
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rab 2800  df-v 3068  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3758  df-if 3909  df-sn 3996  df-pr 3998  df-op 4002  df-br 4435  df-opab 4494  df-id 4795  df-cnv 4888  df-co 4889  df-fun 5635
This theorem is referenced by:  dffun5  5646  dffun6f  5647  sbcfung  5656  dffv2  6005
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