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Theorem dffun6 6557
Description: Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2138, ax-12 2172. (Revised by SN, 19-Dec-2024.)
Assertion
Ref Expression
dffun6 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem dffun6
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dffun2 6554 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2 breq2 5153 . . . . 5 (𝑦 = 𝑧 → (𝑥𝐹𝑦𝑥𝐹𝑧))
32mo4 2561 . . . 4 (∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
43albii 1822 . . 3 (∀𝑥∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
54anbi2i 624 . 2 ((Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦) ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)))
61, 5bitr4i 278 1 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  ∃*wmo 2533   class class class wbr 5149  Rel wrel 5682  Fun wfun 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546
This theorem is referenced by:  dffun3  6558  funmo  6564  funmoOLD  6565  dffun7  6576  fununfun  6597  funcnvsn  6599  funcnv2  6617  svrelfun  6621  funimaexg  6635  fnres  6678  nfunsn  6934  dff3  7102  brdom3  10523  nqerf  10925  shftfn  15020  cnextfun  23568  perfdvf  25420  taylf  25873  funressnvmo  45755
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