ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dffun9 GIF version

Theorem dffun9 4958
Description: Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
dffun9 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dffun9
StepHypRef Expression
1 dffun7 4956 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
2 vex 2577 . . . . . . . 8 𝑥 ∈ V
3 vex 2577 . . . . . . . 8 𝑦 ∈ V
42, 3brelrn 4595 . . . . . . 7 (𝑥𝐴𝑦𝑦 ∈ ran 𝐴)
54pm4.71ri 378 . . . . . 6 (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴𝑥𝐴𝑦))
65mobii 1953 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴𝑥𝐴𝑦))
7 df-rmo 2331 . . . . 5 (∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴𝑥𝐴𝑦))
86, 7bitr4i 180 . . . 4 (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)
98ralbii 2347 . . 3 (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)
109anbi2i 438 . 2 ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))
111, 10bitri 177 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wcel 1409  ∃*wmo 1917  wral 2323  ∃*wrmo 2326   class class class wbr 3792  dom cdm 4373  ran crn 4374  Rel wrel 4378  Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rmo 2331  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator