MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  feu Structured version   Visualization version   GIF version

Theorem feu 6773
Description: There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
Assertion
Ref Expression
feu ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶

Proof of Theorem feu
StepHypRef Expression
1 ffn 6723 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fneu2 6666 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
31, 2sylan 578 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
4 opelf 6758 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → (𝐶𝐴𝑦𝐵))
54simprd 494 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
65ex 411 . . . . . 6 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹𝑦𝐵))
76pm4.71rd 561 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ (𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
87eubidv 2574 . . . 4 (𝐹:𝐴𝐵 → (∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
98adantr 479 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
103, 9mpbid 231 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
11 df-reu 3364 . 2 (∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
1210, 11sylibr 233 1 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  ∃!weu 2556  ∃!wreu 3361  cop 4636   Fn wfn 6544  wf 6545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-fun 6551  df-fn 6552  df-f 6553
This theorem is referenced by:  fdmeu  6954  fsn  7144  f1ofveu  7413
  Copyright terms: Public domain W3C validator