Theorem dffun8 5029

Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5028. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
dffun8	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dffun8

Step	Hyp	Ref	Expression
1		dffun7 5028	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
2		df-mo 1952	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦))
3		vex 2622	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	eldm 4621	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
5		pm5.5 240	. . . . . 6 ⊢ (∃𝑦 𝑥𝐴𝑦 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦))
6	4, 5	sylbi 119	. . . . 5 ⊢ (𝑥 ∈ dom 𝐴 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦))
7	2, 6	syl5bb 190	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 → (∃*𝑦 𝑥𝐴𝑦 ↔ ∃!𝑦 𝑥𝐴𝑦))
8	7	ralbiia 2392	. . 3 ⊢ (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦)
9	8	anbi2i 445	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
10	1, 9	bitri 182	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∃wex 1426   ∈ wcel 1438  ∃!weu 1948  ∃*wmo 1949  ∀wral 2359   class class class wbr 3837  dom cdm 4428  Rel wrel 4433  Fun wfun 4996

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-cnv 4436  df-co 4437  df-dm 4438  df-fun 5004

This theorem is referenced by:  funco  5040  funimaexglem  5083  funfveu  5302