Theorem nfunsn 6870

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 2575	. . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2		vex 3442	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	2	brresi 5944	. . . . . . . . 9 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))
4		velsn 4593	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5		breq1 5098	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
6	4, 5	sylbi 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
7	6	biimpa 476	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) → 𝐴𝐹𝑦)
8	3, 7	sylbi 217	. . . . . . . 8 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 → 𝐴𝐹𝑦)
9	8	moimi 2542	. . . . . . 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 6818	. . . . . 6 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅)
12	10, 11	nsyl4 158	. . . . 5 ⊢ (¬ (𝐹‘𝐴) = ∅ → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
13	12	alrimiv 1928	. . . 4 ⊢ (¬ (𝐹‘𝐴) = ∅ → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
14		relres 5961	. . . 4 ⊢ Rel (𝐹 ↾ {𝐴})
15	13, 14	jctil 519	. . 3 ⊢ (¬ (𝐹‘𝐴) = ∅ → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
16		dffun6 6500	. . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
17	15, 16	sylibr 234	. 2 ⊢ (¬ (𝐹‘𝐴) = ∅ → Fun (𝐹 ↾ {𝐴}))
18	17	con1i 147	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∃*wmo 2535  ∃!weu 2565  ∅c0 4284  {csn 4577   class class class wbr 5095   ↾ cres 5623  Rel wrel 5626  Fun wfun 6483  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-res 5633  df-iota 6445  df-fun 6491  df-fv 6497

This theorem is referenced by:  fvfundmfvn0  6871  dffv2  6926  afv2ndeffv0  47374