Theorem fdmeu 5722

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 5551	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝑋, 𝑦⟩ ∈ 𝐹)
2		ffn 5510	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3	2	anim1i 340	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴))
4	3	adantr 276	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴))
5		fnopfvb 5718	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ 𝐹))
6	4, 5	syl 14	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑋) = 𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ 𝐹))
7	6	reubidva 2730	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦 ↔ ∃!𝑦 ∈ 𝐵 ⟨𝑋, 𝑦⟩ ∈ 𝐹))
8	1, 7	mpbird 167	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∃!wreu 2524  ⟨cop 3694   Fn wfn 5349  ⟶wf 5350  ‘cfv 5354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362

This theorem is referenced by:  uspgriedgedg  16191