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Theorem feu 5978
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 5944 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fneu2 5896 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
31, 2sylan 486 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
4 opelf 5964 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → (𝐶𝐴𝑦𝐵))
54simprd 477 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
65ex 448 . . . . . 6 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹𝑦𝐵))
76pm4.71rd 664 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ (𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
87eubidv 2477 . . . 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 220 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
11 df-reu 2902 . 2 (∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
1210, 11sylibr 222 1 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  ∃!weu 2457  ∃!wreu 2897  cop 4130   Fn wfn 5785  wf 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-fun 5792  df-fn 5793  df-f 5794
This theorem is referenced by:  fsn  6293  f1ofveu  6522
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