Theorem fdmeu 6899

Description: There is exactly one codomain element for each element of the domain of a function. (Contributed by AV, 20-Apr-2025.)

Assertion

Ref	Expression
fdmeu	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦)

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

Proof of Theorem fdmeu

Step	Hyp	Ref	Expression
1		feu 6718	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝑋, 𝑦⟩ ∈ 𝐹)
2		ffn 6670	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3	2	anim1i 615	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴))
4	3	adantr 480	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴))
5		fnopfvb 6894	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ 𝐹))
6	4, 5	syl 17	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑋) = 𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ 𝐹))
7	6	reubidva 3367	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦 ↔ ∃!𝑦 ∈ 𝐵 ⟨𝑋, 𝑦⟩ ∈ 𝐹))
8	1, 7	mpbird 257	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!wreu 3349  ⟨cop 4591   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507

This theorem is referenced by:  uspgriedgedg  29079