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Theorem dffun3f 45294
 Description: Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)
Hypotheses
Ref Expression
dffun3f.1 𝑥𝐴
dffun3f.2 𝑦𝐴
dffun3f.3 𝑧𝐴
Assertion
Ref Expression
dffun3f (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem dffun3f
StepHypRef Expression
1 dffun3f.1 . . 3 𝑥𝐴
2 dffun3f.2 . . 3 𝑦𝐴
31, 2dffun6f 6343 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4 nfcv 2955 . . . . . 6 𝑧𝑥
5 dffun3f.3 . . . . . 6 𝑧𝐴
6 nfcv 2955 . . . . . 6 𝑧𝑦
74, 5, 6nfbr 5080 . . . . 5 𝑧 𝑥𝐴𝑦
87mof 2622 . . . 4 (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
98albii 1821 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
109anbi2i 625 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
113, 10bitri 278 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  ∃*wmo 2596  Ⅎwnfc 2936   class class class wbr 5033  Rel wrel 5527  Fun wfun 6323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-br 5034  df-opab 5096  df-id 5428  df-cnv 5530  df-co 5531  df-fun 6331 This theorem is referenced by:  setrec2lem2  45306
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