Theorem dffun8 6564

Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6563. (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 6563	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
2		moeu 2611	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦))
3		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	eldm 5890	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
5		pm5.5 364	. . . . . 6 ⊢ (∃𝑦 𝑥𝐴𝑦 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦))
6	4, 5	sylbi 220	. . . . 5 ⊢ (𝑥 ∈ dom 𝐴 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦))
7	2, 6	bitrid 286	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 → (∃*𝑦 𝑥𝐴𝑦 ↔ ∃!𝑦 𝑥𝐴𝑦))
8	7	ralbiia 3109	. . 3 ⊢ (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦)
9	8	anbi2i 634	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
10	1, 9	bitri 278	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1809   ∈ wcel 2143  ∃*wmo 2565  ∃!weu 2596  ∀wral 3079   class class class wbr 5109  dom cdm 5661  Rel wrel 5666  Fun wfun 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-fun 6538

This theorem is referenced by:  dfdfat2  47865