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

Theorem eufnfv 5723
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 5719 . . . 4 (𝑥𝐴𝐵) ∈ V
3 eqeq2 2180 . . . . . 6 (𝑦 = (𝑥𝐴𝐵) → (𝑓 = 𝑦𝑓 = (𝑥𝐴𝐵)))
43bibi2d 231 . . . . 5 (𝑦 = (𝑥𝐴𝐵) → (((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑦) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
54albidv 1817 . . . 4 (𝑦 = (𝑥𝐴𝐵) → (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑦) ↔ ∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
62, 5spcev 2825 . . 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 2170 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
97, 8fnmpti 5324 . . . . . 6 (𝑥𝐴𝐵) Fn 𝐴
10 fneq1 5284 . . . . . 6 (𝑓 = (𝑥𝐴𝐵) → (𝑓 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
119, 10mpbiri 167 . . . . 5 (𝑓 = (𝑥𝐴𝐵) → 𝑓 Fn 𝐴)
1211pm4.71ri 390 . . . 4 (𝑓 = (𝑥𝐴𝐵) ↔ (𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)))
13 dffn5im 5540 . . . . . . 7 (𝑓 Fn 𝐴𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
1413eqeq1d 2179 . . . . . 6 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵)))
15 funfvex 5511 . . . . . . . . 9 ((Fun 𝑓𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ V)
1615funfni 5296 . . . . . . . 8 ((𝑓 Fn 𝐴𝑥𝐴) → (𝑓𝑥) ∈ V)
1716ralrimiva 2543 . . . . . . 7 (𝑓 Fn 𝐴 → ∀𝑥𝐴 (𝑓𝑥) ∈ V)
18 mpteqb 5584 . . . . . . 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 187 . . . . 5 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2120pm5.32i 451 . . . 4 ((𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2212, 21bitr2i 184 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))
236, 22mpg 1444 . 2 𝑦𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑦)
24 df-eu 2022 . 2 (∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ ∃𝑦𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑦))
2523, 24mpbir 145 1 ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1346   = wceq 1348  wex 1485  ∃!weu 2019  wcel 2141  wral 2448  Vcvv 2730  cmpt 4048   Fn wfn 5191  cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator