Theorem funeu 5243

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 5235	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		releldm 4864	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
3	1, 2	sylan 283	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
4		eldmg 4824	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (𝐴 ∈ dom 𝐹 ↔ ∃𝑦 𝐴𝐹𝑦))
5	4	ibi 176	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → ∃𝑦 𝐴𝐹𝑦)
6	3, 5	syl 14	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃𝑦 𝐴𝐹𝑦)
7		funmo 5233	. . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
8	7	adantr 276	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
9		df-mo 2030	. . 3 ⊢ (∃*𝑦 𝐴𝐹𝑦 ↔ (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
10	8, 9	sylib 122	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
11	6, 10	mpd 13	1 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1492  ∃!weu 2026  ∃*wmo 2027   ∈ wcel 2148   class class class wbr 4005  dom cdm 4628  Rel wrel 4633  Fun wfun 5212

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-fun 5220

This theorem is referenced by:  funeu2  5244  funbrfv  5556