Theorem fdmeu 6873

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 6694	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝑋, 𝑦⟩ ∈ 𝐹)
2		ffn 6646	. . . . . 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 6868	. . . 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 3360	. 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 1541   ∈ wcel 2111  ∃!wreu 3344  ⟨cop 4577   Fn wfn 6471  ⟶wf 6472  ‘cfv 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484

This theorem is referenced by:  uspgriedgedg  29149