Theorem dffun8 6377

Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6376. (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 6376	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
2		moeu 2664	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦))
3		vex 3497	. . . . . . 7 ⊢ 𝑥 ∈ V
4	3	eldm 5763	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
5		pm5.5 364	. . . . . 6 ⊢ (∃𝑦 𝑥𝐴𝑦 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦))
6	4, 5	sylbi 219	. . . . 5 ⊢ (𝑥 ∈ dom 𝐴 → ((∃𝑦 𝑥𝐴𝑦 → ∃!𝑦 𝑥𝐴𝑦) ↔ ∃!𝑦 𝑥𝐴𝑦))
7	2, 6	syl5bb 285	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 → (∃*𝑦 𝑥𝐴𝑦 ↔ ∃!𝑦 𝑥𝐴𝑦))
8	7	ralbiia 3164	. . 3 ⊢ (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦)
9	8	anbi2i 624	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
10	1, 9	bitri 277	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1776   ∈ wcel 2110  ∃*wmo 2616  ∃!weu 2649  ∀wral 3138   class class class wbr 5058  dom cdm 5549  Rel wrel 5554  Fun wfun 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-id 5454  df-cnv 5557  df-co 5558  df-dm 5559  df-fun 6351

This theorem is referenced by:  dfdfat2  43321