Theorem feu 5300

Description: There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)

Assertion

Ref	Expression
feu	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹)

Distinct variable groups:   𝑦,𝐹   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶

Proof of Theorem feu

Step	Hyp	Ref	Expression
1		ffn 5267	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		fneu2 5223	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹)
3	1, 2	sylan 281	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹)
4		opelf 5289	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
5	4	simprd 113	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ 𝐵)
6	5	ex 114	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ 𝐵))
7	6	pm4.71rd 391	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
8	7	eubidv 2005	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
9	8	adantr 274	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
10	3, 9	mpbid 146	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
11		df-reu 2421	. 2 ⊢ (∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
12	10, 11	sylibr 133	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1480  ∃!weu 1997  ∃!wreu 2416  ⟨cop 3525   Fn wfn 5113  ⟶wf 5114

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122

This theorem is referenced by:  fsn  5585  f1ofveu  5755