Theorem nfunsn 5228

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 1973	. . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2		vex 2604	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	2	brres 4636	. . . . . . . . 9 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}))
4		velsn 3415	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5		breq1 3788	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
6	4, 5	sylbi 119	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
7	6	biimpac 292	. . . . . . . . 9 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}) → 𝐴𝐹𝑦)
8	3, 7	sylbi 119	. . . . . . . 8 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 → 𝐴𝐹𝑦)
9	8	moimi 2006	. . . . . . 7 ⊢ (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
10	1, 9	syl 14	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
11	10	alrimiv 1795	. . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
12		relres 4657	. . . . 5 ⊢ Rel (𝐹 ↾ {𝐴})
13	11, 12	jctil 305	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
14		dffun6 4936	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
15	13, 14	sylibr 132	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
16	15	con3i 594	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ ∃!𝑦 𝐴𝐹𝑦)
17		tz6.12-2 5189	. 2 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅)
18	16, 17	syl 14	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   = wceq 1284   ∈ wcel 1433  ∃!weu 1941  ∃*wmo 1942  ∅c0 3251  {csn 3398   class class class wbr 3785   ↾ cres 4365  Rel wrel 4368  Fun wfun 4916  ‘cfv 4922

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-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-res 4375  df-iota 4887  df-fun 4924  df-fv 4930

This theorem is referenced by: (None)