Theorem funfveu 5568

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 2256	. . . . 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 4033	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
4	3	eubidv 2050	. . . 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 5283	. . . . 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 2582	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦)
9	5, 8	vtoclg 2821	. 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦))
10	9	anabsi7 581	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃!weu 2042   ∈ wcel 2164  ∀wral 2472   class class class wbr 4030  dom cdm 4660  Rel wrel 4665  Fun wfun 5249

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-id 4325  df-cnv 4668  df-co 4669  df-dm 4670  df-fun 5257

This theorem is referenced by:  funfvex  5572