Theorem nfunsn 6943

Description: If the restriction of a class to a singleton is not a function, then its value is the empty set. (An artifact of our function value definition.) (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 2567	. . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2		vex 3466	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	2	brresi 5998	. . . . . . . . 9 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))
4		velsn 4649	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5		breq1 5156	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
6	4, 5	sylbi 216	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
7	6	biimpa 475	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) → 𝐴𝐹𝑦)
8	3, 7	sylbi 216	. . . . . . . 8 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 → 𝐴𝐹𝑦)
9	8	moimi 2534	. . . . . . 7 ⊢ (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
10	1, 9	syl 17	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
11		tz6.12-2 6889	. . . . . 6 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅)
12	10, 11	nsyl4 158	. . . . 5 ⊢ (¬ (𝐹‘𝐴) = ∅ → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
13	12	alrimiv 1923	. . . 4 ⊢ (¬ (𝐹‘𝐴) = ∅ → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
14		relres 6015	. . . 4 ⊢ Rel (𝐹 ↾ {𝐴})
15	13, 14	jctil 518	. . 3 ⊢ (¬ (𝐹‘𝐴) = ∅ → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
16		dffun6 6567	. . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
17	15, 16	sylibr 233	. 2 ⊢ (¬ (𝐹‘𝐴) = ∅ → Fun (𝐹 ↾ {𝐴}))
18	17	con1i 147	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∃*wmo 2527  ∃!weu 2557  ∅c0 4325  {csn 4633   class class class wbr 5153   ↾ cres 5684  Rel wrel 5687  Fun wfun 6548  ‘cfv 6554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-res 5694  df-iota 6506  df-fun 6556  df-fv 6562

This theorem is referenced by:  fvfundmfvn0  6944  dffv2  6997  afv2ndeffv0  46873