Theorem feu 6262

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 6223	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		fneu2 6174	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹)
3	1, 2	sylan 575	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹)
4		opelf 6247	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
5	4	simprd 489	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ 𝐵)
6	5	ex 401	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ 𝐵))
7	6	pm4.71rd 558	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
8	7	eubidv 2585	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
9	8	adantr 472	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
10	3, 9	mpbid 223	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
11		df-reu 3062	. 2 ⊢ (∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
12	10, 11	sylibr 225	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∈ wcel 2155  ∃!weu 2581  ∃!wreu 3057  ⟨cop 4340   Fn wfn 6063  ⟶wf 6064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-fun 6070  df-fn 6071  df-f 6072

This theorem is referenced by:  fsn  6593  f1ofveu  6837