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: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 6723	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		fneu2 6666	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹)
3	1, 2	sylan 578	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹)
4		opelf 6758	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
5	4	simprd 494	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ 𝐵)
6	5	ex 411	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ 𝐵))
7	6	pm4.71rd 561	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
8	7	eubidv 2574	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
9	8	adantr 479	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (∃!𝑦⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
10	3, 9	mpbid 231	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
11		df-reu 3364	. 2 ⊢ (∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
12	10, 11	sylibr 233	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹)

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