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Theorem dffn5im 5246
Description: Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 4965 and dmmptss 4844. (Contributed by Jim Kingdon, 31-Dec-2018.)
Assertion
Ref Expression
dffn5im (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dffn5im
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fnrel 5024 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 dfrel4v 4799 . . . 4 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
31, 2sylib 131 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4 fnbr 5028 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
54ex 112 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
65pm4.71rd 380 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
7 eqcom 2058 . . . . . . 7 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
8 fnbrfvb 5241 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
97, 8syl5bb 185 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
109pm5.32da 433 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
116, 10bitr4d 184 . . . 4 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹𝑥))))
1211opabbidv 3850 . . 3 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
133, 12eqtrd 2088 . 2 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
14 df-mpt 3847 . 2 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
1513, 14syl6eqr 2106 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409   class class class wbr 3791  {copab 3844  cmpt 3845  Rel wrel 4377   Fn wfn 4924  cfv 4929
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 3902  ax-pow 3954  ax-pr 3971
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-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937
This theorem is referenced by:  fnrnfv  5247  feqmptd  5253  dffn5imf  5255  eqfnfv  5292  fndmin  5301  fcompt  5360  resfunexg  5409  eufnfv  5416  fnovim  5636  offveqb  5757  caofinvl  5760  oprabco  5865  df1st2  5867  df2nd2  5868
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