Theorem nfunsn 5520

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 2046	. . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2		vex 2729	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	2	brres 4890	. . . . . . . . 9 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}))
4		velsn 3593	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5		breq1 3985	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
6	4, 5	sylbi 120	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
7	6	biimpac 296	. . . . . . . . 9 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}) → 𝐴𝐹𝑦)
8	3, 7	sylbi 120	. . . . . . . 8 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 → 𝐴𝐹𝑦)
9	8	moimi 2079	. . . . . . 7 ⊢ (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
10	1, 9	syl 14	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
11	10	alrimiv 1862	. . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
12		relres 4912	. . . . 5 ⊢ Rel (𝐹 ↾ {𝐴})
13	11, 12	jctil 310	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
14		dffun6 5202	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
15	13, 14	sylibr 133	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
16	15	con3i 622	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ ∃!𝑦 𝐴𝐹𝑦)
17		tz6.12-2 5477	. 2 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅)
18	16, 17	syl 14	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃!weu 2014  ∃*wmo 2015   ∈ wcel 2136  ∅c0 3409  {csn 3576   class class class wbr 3982   ↾ cres 4606  Rel wrel 4609  Fun wfun 5182  ‘cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-res 4616  df-iota 5153  df-fun 5190  df-fv 5196

This theorem is referenced by: (None)