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Theorem dffun3f 43324
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 6137 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4 nfcv 2969 . . . . . 6 𝑧𝑥
5 dffun3f.3 . . . . . 6 𝑧𝐴
6 nfcv 2969 . . . . . 6 𝑧𝑦
74, 5, 6nfbr 4920 . . . . 5 𝑧 𝑥𝐴𝑦
87mof 2631 . . . 4 (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
98albii 1920 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧))
109anbi2i 618 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
113, 10bitri 267 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1656  wex 1880  ∃*wmo 2603  wnfc 2956   class class class wbr 4873  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:  setrec2lem2  43336
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