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Theorem eufnfv 5442
Description: A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
Hypotheses
Ref Expression
eufnfv.1 𝐴 ∈ V
eufnfv.2 𝐵 ∈ V
Assertion
Ref Expression
eufnfv ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝑓,𝐴   𝐵,𝑓
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eufnfv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eufnfv.1 . . . . 5 𝐴 ∈ V
21mptex 5440 . . . 4 (𝑥𝐴𝐵) ∈ V
3 eqeq2 2092 . . . . . 6 (𝑦 = (𝑥𝐴𝐵) → (𝑓 = 𝑦𝑓 = (𝑥𝐴𝐵)))
43bibi2d 230 . . . . 5 (𝑦 = (𝑥𝐴𝐵) → (((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑦) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
54albidv 1747 . . . 4 (𝑦 = (𝑥𝐴𝐵) → (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑦) ↔ ∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
62, 5spcev 2701 . . 3 (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵)) → ∃𝑦𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑦))
7 eufnfv.2 . . . . . . 7 𝐵 ∈ V
8 eqid 2083 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
97, 8fnmpti 5078 . . . . . 6 (𝑥𝐴𝐵) Fn 𝐴
10 fneq1 5038 . . . . . 6 (𝑓 = (𝑥𝐴𝐵) → (𝑓 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
119, 10mpbiri 166 . . . . 5 (𝑓 = (𝑥𝐴𝐵) → 𝑓 Fn 𝐴)
1211pm4.71ri 384 . . . 4 (𝑓 = (𝑥𝐴𝐵) ↔ (𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)))
13 dffn5im 5272 . . . . . . 7 (𝑓 Fn 𝐴𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
1413eqeq1d 2091 . . . . . 6 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵)))
15 funfvex 5244 . . . . . . . . 9 ((Fun 𝑓𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ V)
1615funfni 5050 . . . . . . . 8 ((𝑓 Fn 𝐴𝑥𝐴) → (𝑓𝑥) ∈ V)
1716ralrimiva 2439 . . . . . . 7 (𝑓 Fn 𝐴 → ∀𝑥𝐴 (𝑓𝑥) ∈ V)
18 mpteqb 5314 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ V → ((𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
1917, 18syl 14 . . . . . 6 (𝑓 Fn 𝐴 → ((𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2014, 19bitrd 186 . . . . 5 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2120pm5.32i 442 . . . 4 ((𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2212, 21bitr2i 183 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))
236, 22mpg 1381 . 2 𝑦𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑦)
24 df-eu 1946 . 2 (∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ ∃𝑦𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑦))
2523, 24mpbir 144 1 ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434  ∃!weu 1943  wral 2353  Vcvv 2610  cmpt 3859   Fn wfn 4947  cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960
This theorem is referenced by: (None)
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