Theorem funeu 6603

Description: There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
funeu	⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem funeu

Step	Hyp	Ref	Expression
1		funrel 6595	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		releldm 5969	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
3	1, 2	sylan 579	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
4		eldmg 5923	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (𝐴 ∈ dom 𝐹 ↔ ∃𝑦 𝐴𝐹𝑦))
5	4	ibi 267	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → ∃𝑦 𝐴𝐹𝑦)
6	3, 5	syl 17	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃𝑦 𝐴𝐹𝑦)
7		funmo 6593	. . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
8	7	adantr 480	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
9		moeu 2586	. . 3 ⊢ (∃*𝑦 𝐴𝐹𝑦 ↔ (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
10	8, 9	sylib 218	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
11	6, 10	mpd 15	1 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  ∃*wmo 2541  ∃!weu 2571   class class class wbr 5166  dom cdm 5700  Rel wrel 5705  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-fun 6575

This theorem is referenced by:  funeu2  6604  funbrfv  6971  frege124d  43723  funbrafv2  47162