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Theorem dffun8 6125
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6124. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
dffun8 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dffun8
StepHypRef Expression
1 dffun7 6124 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
2 df-mo 2635 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦))
3 vex 3394 . . . . . . 7 𝑥 ∈ V
43eldm 5522 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
5 pm5.5 352 . . . . . 6 (∃𝑦 𝑥𝐴𝑦 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦))
64, 5sylbi 208 . . . . 5 (𝑥 ∈ dom 𝐴 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦))
72, 6syl5bb 274 . . . 4 (𝑥 ∈ dom 𝐴 → (∃*𝑦 𝑥𝐴𝑦 ↔ ∃!𝑦 𝑥𝐴𝑦))
87ralbiia 3167 . . 3 (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦)
98anbi2i 611 . 2 ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
101, 9bitri 266 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wex 1859  wcel 2156  ∃!weu 2630  ∃*wmo 2631  wral 3096   class class class wbr 4844  dom cdm 5311  Rel wrel 5316  Fun wfun 6091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-id 5219  df-cnv 5319  df-co 5320  df-dm 5321  df-fun 6099
This theorem is referenced by:  dfdfat2  41711
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