Theorem funfveu 5661

Description: A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
funfveu	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem funfveu

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2294	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹))
2	1	anbi2d 464	. . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)))
3		breq1 4096	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
4	3	eubidv 2087	. . . 4 ⊢ (𝑥 = 𝐴 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
5	2, 4	imbi12d 234	. . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)))
6		dffun8 5361	. . . . 5 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦))
7	6	simprbi 275	. . . 4 ⊢ (Fun 𝐹 → ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦)
8	7	r19.21bi 2621	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦)
9	5, 8	vtoclg 2865	. 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦))
10	9	anabsi7 583	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃!weu 2079   ∈ wcel 2202  ∀wral 2511   class class class wbr 4093  dom cdm 4731  Rel wrel 4736  Fun wfun 5327

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-cnv 4739  df-co 4740  df-dm 4741  df-fun 5335

This theorem is referenced by:  funfvex  5665