Theorem fdmeu 6941

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 6758	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝑋, 𝑦⟩ ∈ 𝐹)
2		ffn 6709	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3	2	anim1i 627	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴))
4	3	adantr 486	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴))
5		fnopfvb 6936	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ 𝐹))
6	4, 5	syl 18	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑋) = 𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ 𝐹))
7	6	reubidva 3385	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦 ↔ ∃!𝑦 ∈ 𝐵 ⟨𝑋, 𝑦⟩ ∈ 𝐹))
8	1, 7	mpbird 260	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃!wreu 3369  ⟨cop 4597   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548

This theorem is used by:  uspgriedgedg  29560