Theorem nfunsn 5610

Description: If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
nfunsn	⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)

Proof of Theorem nfunsn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eumo 2085	. . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2		vex 2774	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	2	brres 4964	. . . . . . . . 9 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}))
4		velsn 3649	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5		breq1 4046	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
6	4, 5	sylbi 121	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
7	6	biimpac 298	. . . . . . . . 9 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}) → 𝐴𝐹𝑦)
8	3, 7	sylbi 121	. . . . . . . 8 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 → 𝐴𝐹𝑦)
9	8	moimi 2118	. . . . . . 7 ⊢ (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
10	1, 9	syl 14	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
11	10	alrimiv 1896	. . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
12		relres 4986	. . . . 5 ⊢ Rel (𝐹 ↾ {𝐴})
13	11, 12	jctil 312	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
14		dffun6 5284	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
15	13, 14	sylibr 134	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
16	15	con3i 633	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ ∃!𝑦 𝐴𝐹𝑦)
17		tz6.12-2 5566	. 2 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅)
18	16, 17	syl 14	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1370   = wceq 1372  ∃!weu 2053  ∃*wmo 2054   ∈ wcel 2175  ∅c0 3459  {csn 3632   class class class wbr 4043   ↾ cres 4676  Rel wrel 4679  Fun wfun 5264  ‘cfv 5270

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-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-res 4686  df-iota 5231  df-fun 5272  df-fv 5278

This theorem is referenced by: (None)