Theorem fdmeu 6965

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 6785	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝑋, 𝑦⟩ ∈ 𝐹)
2		ffn 6737	. . . . . 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 6961	. . . 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 3394	. 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 1537   ∈ wcel 2106  ∃!wreu 3376  ⟨cop 4637   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571

This theorem is referenced by:  uspgriedgedg  29208