Theorem funeu 5379

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 5371	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		releldm 4994	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
3	1, 2	sylan 283	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
4		eldmg 4953	. . . 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 5369	. . . 4 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
8	7	adantr 276	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
9		df-mo 2086	. . 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 1541  ∃!weu 2082  ∃*wmo 2083   ∈ wcel 2205   class class class wbr 4111  dom cdm 4751  Rel wrel 4756  Fun wfun 5348

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-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  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-fun 5356

This theorem is referenced by:  funeu2  5380  funbrfv  5715