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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:𝐴𝐵𝑋𝐴) → ∃!𝑦𝐵 (𝐹𝑋) = 𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐹   𝑦,𝑋

Proof of Theorem fdmeu
StepHypRef Expression
1 feu 6784 . 2 ((𝐹:𝐴𝐵𝑋𝐴) → ∃!𝑦𝐵𝑋, 𝑦⟩ ∈ 𝐹)
2 ffn 6736 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
32anim1i 615 . . . . 5 ((𝐹:𝐴𝐵𝑋𝐴) → (𝐹 Fn 𝐴𝑋𝐴))
43adantr 480 . . . 4 (((𝐹:𝐴𝐵𝑋𝐴) ∧ 𝑦𝐵) → (𝐹 Fn 𝐴𝑋𝐴))
5 fnopfvb 6960 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = 𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ 𝐹))
64, 5syl 17 . . 3 (((𝐹:𝐴𝐵𝑋𝐴) ∧ 𝑦𝐵) → ((𝐹𝑋) = 𝑦 ↔ ⟨𝑋, 𝑦⟩ ∈ 𝐹))
76reubidva 3396 . 2 ((𝐹:𝐴𝐵𝑋𝐴) → (∃!𝑦𝐵 (𝐹𝑋) = 𝑦 ↔ ∃!𝑦𝐵𝑋, 𝑦⟩ ∈ 𝐹))
81, 7mpbird 257 1 ((𝐹:𝐴𝐵𝑋𝐴) → ∃!𝑦𝐵 (𝐹𝑋) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  ∃!wreu 3378  cop 4632   Fn wfn 6556  wf 6557  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  uspgriedgedg  29193
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